Harmonic Analysis and Representation Theory

of the Automorphism Groups of Homogeneous Trees

An Honours Thesis

Corran Webster

Supervised by Prof. Sutherland

November 8, 1991

Contents

Acknowledgements iii

Introduction 1

1 Automorphisms of Trees 2 1.1 Graphs and Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Concepts from Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Metrics on Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 The Boundary of an Infinite Tree . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Automorphisms of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 The Automorphism Group of a Tree . . . . . . . . . . . . . . . . . . . . . . 21 1.7 Point Stabiliser Subgroups, K_{v} . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.8 Faithful Transitive Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.9 Extensions and Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Representations of Groups on a Tree 31 2.1 Haar Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Representations of Aut(Γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Special Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Cuspidal Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.6 Extensions and Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . 53

Bibliography 55

List of Figures

1.1 Homogeneous trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Semi-homogeneous trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Barycentric Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 An Open Neighbourhood of ω . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 “Pruning” Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Diagram for Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.7 Restrictions of K_{o}^{r} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 The Orientation E_{+} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Acknowledgements

I would like to thank all those people who have helped me produce this thesis. First and foremost, of course, is my supervisor, Colin Sutherland, who put up with me for most of the year, and answered what must have seemed to be some stupid questions, whilst remaining in good humour throughout. Secondly, I would like to thank those other lecturers who provided help, inspiration or insight—most notably Tony Dooley, Ian Doust and Rod James.

On the typesetting front, I would like to thank Fiona Thorne, for providing me with a copy of OzTEXwhen it became apparent that Microsoft Word was not up to the task, and Daniel, Guy and Ian Doust, for occasional help in wrestling with TEXand L^{A}TEX. (incidentally \mbox{\rm \O} produces nice empty set symbols).

Most importantly, I must thank my family who not only have put up with a computer room with an ever increasing entropy (as I write, manuscripts cover every available horizontal surface) and excessive use of computer time at home, but who also helped with the proofreading of the thesis.

Finally, I would like to thank all the 3rd and 4th year students, and the MathSoc excecutive, for keeping me sort of sane throughout the year.

Corran Webster, 7/11/91

This thesis was produced on a Macintosh IIsi running OzTEX, and using the Macintosh Programmer’s Workshop (MPW) as a text editor.

Introduction

This thesis aims to develop some theory about representation theory and harmonic analysis on the automorphism group of a homogeneous tree. To this end, it is divided up into two chapters.

The first details the graph theory, group theory and geometry associated with homogeneous trees that is required for the second chapter. Although trees occur in many areas of mathematics, the first study of homogeneous trees was instituted by Jean-Pierre Serre, and his book Trees [11] was the outcome of this. These ideas were added to by P. Cartier, amongst others. Underlying the geometric and graph theoretic aspects, are some concepts from group theory which are not part of a usual introductory course on group theory, so for this reason I have included a section, largely derived from Robinson [10] and Serre [11].

The second chapter starts with some introductory theory on Haar Measures and Representations, and then classifies and discusses two specific types of representations on the automorphism group of a homogeneous tree. The theory of the special and cuspidal representations is essentially due to G. I. Ol’shianskii, with some refinements due to Allessandro Figà-Talamanca and Claudio Nebbia. However in the treatment of the cuspidal representations I have attempted to highlight the similarities between these and the special representations, at the cost of glossing over some of the technical details.

At the end of each chapter I have included a short section which gives some additional information, without proof, about how the substance of the chapter might be generalised, or important extensions to the theory.

The source for most of this thesis has been A. Figà-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees [4], and most of the results and proofs come from here, although many have been altered to some extent. I have been slightly more thourough at the basic levels, providing proofs for a number of results that are simply stated in Figà-Talamanca, and going into more detail in some of the other areas, such as the discussion of the point stabiliser subgroups. The only other major difference is the method of showing unimodularity of the automorphism group. I have approached though the expression of the automorphism group as a product of groups, whilst Figà-Talamanca uses the concept of Gelfand pairs to get the same result.

Chapter 1

Automorphisms of Trees

In this chapter we build up some graph and group theory and then investigate the automorphisms of trees and the structure of the group of automorphisms of a tree. We basically follow Figà-Talamanca [4] throughout this chapter, however Section 1.2 introduces some ideas from group theory which are largely from Robinson [10], and occasional ideas come from Serre [11].

1.1 Graphs and Trees

Before we proceed, we need some basic graph theory.

Definition 1.1.1 We make the following definitions:

An oriented graph Γ = (V, E) consists of a set V, of vertices of Γ, a set E, of (directed) edges of Γ, and an incidence relation which is a map

y ↦ (o(y), t(y)) : E → V × V

For y ∈ Y we call o(y) the origin of y, t(y) the terminus of y, and together they are the extremities of y.

If the incidence relation is injective, then we can denote each y ∈ Y uniquely as an ordered pair (u, v) ∈ V × V, where u = o(y) and v = t(y).

If, in addition, an oriented graph Γ = (V, E) has a map

y ↦ \overline{y} : E → E

such that for each y ∈ Y we have \overline{\overline{y}} = y, \overline{y} ≠ y and o(y) = t(\overline{y}), then Γ is referred to as a (non-directed) graph.

For y ∈ Y we call, \overline{y} the inverse of y, the pair \{y, \overline{y}\} a (non-directed) edge of Γ. Again, we refer to o(y) = t(\overline{y}) and t(y) = o(\overline{y}) as the extremities of the edge \{y, \overline{y}\}.

If the incidence relation is injective, then we can denote every non-directed edge \{y, \overline{y}\} uniquely by the unordered pair {u, v}, where u, v ∈ V and u, v are the extremities of \{y, \overline{y}\}.

Some references refer to graphs where the incidence relation is not injective as multigraphs. Graphs can be represented diagrammatically by marking a point for each vertex of the

graph, and a line for each pair of edges \{y, \overline{y}\}, with endpoints of the line being the points corresponding to the vertices which are the extremities of the edge, like this:

[figure]

In a similar fashion, we can draw a diagram representing a directed graph, but we represent the directed edge y by a line with an arrow pointing from o(y) to t(y), like this:

[figure]

Definition 1.1.2 Let Γ = (V, E) be a (possibly directed) graph.

A pair of distinct vertices are adjacent if they are extremities of the same edge.

The valence or degree of a vertex v ∈ V, val(v), is the number of directed edges y ∈ E for which o(y) = v. If Γ is non-directed, then this equals the number of directed edges y ∈ E for which t(y) = v. The valence of a vertex may be infinite.

A graph is locally finite if the valence of every vertex is finite.

A path in Γ is an alternating sequence of vertices and edges, v_{0}, y_{0}, v_{1}, y_{1}, …, y_{n−1}, v_{n}, where v_{i} ∈ V, y_{i} ∈ E and o(y_{i}) = v_{i} and t(y_{i}) = v_{i+1}, for all i.

The length of a path with n edges, v_{0}, y_{0}, v_{1}, y_{1}, …, y_{n−1}, v_{n}, is n.

A graph Γ is said to be connected if all pairs of distinct vertices have a path between them.

A chain in Γ is a path, v_{0}, y_{0}, v_{1}, y_{1}, …, y_{n−1}, v_{n}, for which y_{i} ≠ \overline{y_{i+1}} for all i ∈ \{0, …, n−2\}.

A loop in Γ is a chain, v_{0}, y_{0}, v_{1}, y_{1}, …, y_{n−1}, v_{n}, where v_{0} = v_{n}

Note that if the incidence relation is injective, then we can refer to a path simply by the vertices, as there is a unique edge defined by the extremities. In this case a path is a chain if v_{i} ≠ v_{i+2}, for all i ∈ {0, …, n − 2}.

We have now reached the point where we can define what a tree actually is.

Definition 1.1.3 A tree is a connected, non-directed graph with no loops.

Figure 1.1: Homogeneous trees

A direct consequence of this definition is that given a tree Γ = (V, E), every pair of points u, v ∈ V has a unique chain between them (existence follows from the connectedness of the graph, and uniqueness from the fact that there are no loops). We call this unique chain the geodesic between u and v, and we represent it symbolically as [u, v].

The following are the two types of trees which will be of special interest.

Definition 1.1.4 A homogeneous tree of order q +1 is a locally finite tree where each vertex has valence q + 1.

A semi-homogeneous tree of order (q + 1, r + 1) is a locally finite tree where each vertex has either a valence of q + 1 or a valence of r + 1, and no two vertices of the same order are adjacent.

(The convention of using q + 1 and r + 1 springs from convenience in handling spherical functions)

One could conceive of a homogeneous tree of order 0 as being a single vertex with no edges, however this is a trivial case and we will ignore it. One must also note that a semihomogeneous tree of order (q + 1, q + 1) is in fact a homogeneous tree of order q + 1. The first few examples of each type of tree are illustrated in figures 1.1 and 1.2.

Having defined the objects with which this thesis is concerned, we will now define some further useful concepts from graph theory:

Figure 1.2: Semi-homogeneous trees 5 Definition 1.1.5 Two graphs Γ = (V, E) and Λ = (U, D) are isomorphic if there exists a bijective map α : Γ → Λ such that α maps vertices to vertices, edges to edges and preserves the incidence relations, ie. if i and j are the respective incidence relations, then i = j ∘ α.

The map α is called a (graph) isomorphism.

Definition 1.1.6 Let Γ = (V, E) be a (possibly directed) graph. (U, D) is a subgraph of Γ if U ⊆ V, D ⊆ E such that for all y ∈ D, o(y) ∈ U and t(y) ∈ U.

A subtree of a graph is a subgraph which is also a tree.

A subtree T = (U, D) is complete if for all u ∈ U, the valence of u in T is either 0, 1 or the same as the valence of u in Γ. (The 0 valence case only applies for the trivial tree U = {u}.)

Note that since a tree has no loops, a subgraph of a tree is automatically a subtree if it is connected and non-directed.

There is nothing unusual about the above definitions, and they are almost exactly what one would expect.

Definition 1.1.7 Let Γ = (V, E) be a non-directed graph.

An orientation of Γ is a subset E_{+} of E, such that E is the disjoint union E_{+} ⊔ \overline{E}_{+}.

An orientation can be considered a way of getting a pair of directed subgraphs from a non-directed graph, by setting Γ_{+} = (V, E_{+}) and \overline{Γ}_{+} = (V, \overline{E}_{+}).

Definition 1.1.8 Let Γ = (V, E) be a (possibly directed) graph. The barycentric subdivision of Γ is the graph Γ′ = (V′, E′) where V′ = V ∪ E and E′ = E × {0, 1}, with the incidence relation

(y, i) ↦ (o′(y, i), t′(y, i))

defined by

o′(y, 0) = o(y)
t′(y, 0) = y = o′(y, 1)
t′(y, 1) = t(y)

Intuitively, the effect of barycentric subdivision is to create a new vertex in the “middle” of each edge. (See figure 1.3).

1.2 Concepts from Group Theory

In this section we will explore some ideas from group theory, essentially following Robinson [10], and make some connections to the theory of graphs.

Definition 1.2.1 Given a group G, and a subset X of G, we define the subgroup generated by X, written < X >, as the subgroup which is the intersection of all subgroups which contain X. We also define the normal closure of X in G to be the normal subgroup which is the intersection of all normal subgroups which contain X.

[figure: Γ; Γ′; Γ′′]

Figure 1.3: Barycentric Subdivision

We notice that < X > is the set of all elements of the form x_{0}^{ε_{0}} … x_{k}^{ε_{k}} where ε_{i} ∈ {−1, +1}, x_{i} ∈ X and k ≥ 0 (if k = 0, then we treat the product as being 1_{G}). This leads us to the following definitions.

Definition 1.2.2 Given an arbitrary set S, called an alphabet of symbols, we define:

A word on S is a finite sequence symbols of S, (s_{1}, …, s_{n}), usually written s_{1} … s_{n}, with s_{i} ∈ S ∪ S^{−1} for 1 ≤ i ≤ n, where S^{−1} is the set {t^{−1} : t ∈ S}.

A reduced word on S is word where no s ∈ S is adjacent to its inverse s^{−1} ∈ S^{−1}, that is, there are no occurrences of ss^{−1} or s^{−1}s for any s ∈ S.

The product of two words s_{1} … s_{n} and t_{1} … t_{m} is the concatenation of the two words, ie. the word s_{1} … s_{n}t_{1} … t_{m}.

The inverse of a word s_{1} … s_{n} is the word written in the reverse order with every term inverted, ie. the word s_{n}^{−1} … s_{1}^{−1} (we define (s^{−1})^{−1} = s).

The empty word e is the word with no symbols.

We note that every word can be turned into a unique reduced word by repeatedly removing all adjacent pairs of symbols and their inverses. We can therefore define concatenation on reduced words by normal concatenation, followed by reduction. It it is easy to see that for set X, the set of reduced words on that set forms a group under this concatenation, with the inverse being the word inverse, and e the identity of the group.

Definition 1.2.3 Given a set X, the group F of reduced words on X is called the free group on X.

We note that F = < X >, and we will often write < X > for the free group on X.

An important result of concerning free groups is that given a set X and a map ϕ from X to some group G, there is a homomorphism α from < X > → G such that given the map β from X into < X > defined by identifying X with the appropriate generating element of < X >, ϕ = α ∘ β. This is called the universal property of free groups.

A corollary of this is that is that every group is the image of a free group under some homomorphism, that is, given a group G, there exists a free group F and an epimorphism (surjective homomorphism) P : F → G. This is derived from the universal property by setting X = G, and getting F = < G >.

Definition 1.2.4 Such an epimorphism P : F → G is called a (free) presentation of G.

The normal subgroup of F, R = ker(P), is called the set of relators of the presentation.

If S is a subset of F such that the normal closure of S in F is R, then we call S a set of defining relators and all r ∈ R \ S are called consequences of S.

Given a projection P : F → G, if F = < X >, then Y = P (X) generates G. If S is a set of defining relators, then we will write G = < X | S >, since G is isomorphic to the group generated by G with the normal closure of S factored out. Sometimes we will simply refer to < X | S > as a presentation of G. We may also write this as < X | s = e : s ∈ S >.

We now use presentations to provide a link between groups and graphs.

Definition 1.2.5 Given a group G with a presentation P such that G =< X | R >, we define the Cayley graph Γ(G, P) of the presentation as the graph Γ = (G, E), where for all g, h ∈ G, (g, h) is an edge if ∃ x ∈ X such that gx = h or gx^{−1} = h.

We notice immediately that the Cayley graph must be connected and that if the number of generators is finite, then the valence of every vertex of Γ must be constant.

Example 1.2.1 The cyclic group C_{4} over the symbol a has a presentation < a | a^{4} = e >. So the edge set of the Cayley graph is:

{{e, a}, {a, a^{2}}, {a^{2}, a^{−1}}, {a^{−1}, e}},

[figure: e; a; a^{2}; a^{−1}]

Example 1.2.2 The group Z has a presentation < 1 >. So the edge set of the Cayley graph is:

{…, {−2, −1}, {−1, 0}, {0, 1}, {1, 2}, …}

[figure: -2; -1; 0; 1; 2]

We note that the Cayley graph of Example 1.2.2 is a tree, and as we are interested in trees, we want to know if there are other groups which have Cayley graphs which are trees.

Proposition 1.2.1 The Cayley graph of a free group, F, is a tree.

Proof:

We know that the Cayley graph of F is connected, so it only remains to be shown that it has no loops.

Assume that there is a loop in the Cayley graph of F. That is, there is a chain γ = x_{0}, …, x_{n}, with x_{0} = x_{n}. This implies that there is a reduced word, W, of generators of F of length n such that x_{0}W = x_{n} = x_{0}, that is W = e. But by the definition F, there is no such word, other than the empty word, e, which has length n = 0, and no chain can have length n = 0. So we have a contradiction.

All these trees have an even valence for their vertices. There are groups which have Cayley graphs which are trees where all the vertices have odd valence, but to describe these we need the concept of a free product.

Definition 1.2.6 Given a family of (disjoint) groups, {G_{i}}_{i∈I} for some index set I, we define the free product of the G_{i} to be

∗_{i∈I} G_{i} = < ⋃_{i∈I} G_{i} | S >

where

S = {g_{i_{1}}g_{i_{2}}g_{i_{3}}^{−1} : g_{i_{1}}, g_{i_{2}} ∈ G_{i} for some i ∈ I, g_{i_{3}} = g_{i_{1}}g_{i_{2}}}

On an intuitive level, this construction is imposing the multiplication of each group on all appropriate elements of the free group generated by the G_{i}, since what we are doing is in effect setting g_{i_{1}}g_{i_{2}}g_{i_{3}}^{−1} = e, ie. g_{i_{1}}g_{i_{2}} = g_{i_{3}}.

More concretely, it can be shown (Robinson [10, §6.2, p163]) that the elements of the free product are exactly the reduced words of the form g_{1}g_{2} … g_{3} where if g_{i} ∈ G_{j}, then g_{i+1} ∉ G_{j} and with multiplication being concatenation of the words, followed by reduction via

If g_{i}, g_{i+1} ∈ G_{j} for some j, then replace them by their product in G_{j}, g_{i}g_{i+1}.

If g_{i} = e_{j}, the identity element of G_{j}, then remove g_{i}.

Repeat these steps as necessary.

Example 1.2.3 Let G_{1} = G_{2} = Z_{2}, then if a is a generator of G_{1}, and b is a generator of G_{2}, then

G_{1} ∗ G_{2} =< {a, b} | a^{2} = b^{2} = e >

or, using the reduced word form,

G_{1} ∗ G_{2} = {…, baba, bab, ba, b, e, a, ab, aba, abab, …}

The next obvious step is to look at the Cayley graph of this group. We get the edge set:

{…, {ba, b}, {b, e}, {e, a}, {a, ab}, …}

which has the following diagram:

[figure: ba; b; e; a; ab]

We note that the graph of this group and that of the integers are isomorphic.

Example 1.2.4 Let G_{i} = Z for i = 1, …, n. Then

∗_{i=1}^{n} Z = < {a_{i}^{j} : i = 1, …, n, j ∈ Z} | {a_{i}^{j}a_{i}^{k}a_{i}^{−j−k} : i = 1, …, n, j, k ∈ Z} >
= < {a_{i} : i = 1, …, n} >
= F_{n}

So the free product of n copies of the integers is F_{n}. In fact F_{n} ∗ F_{m} = F_{n+m}.

We now have enough to describe a large class of groups for which the Cayley graphs are trees.

Theorem 1.2.2 The Cayley graph of

F_{s} ∗ \underbrace{Z_{2} ∗ ⋯ ∗ Z_{2}}_{t times}

is a homogeneous tree of order q + 1 where q + 1 = 2s + t.

Proof:

The graph is a Cayley graph, so it is connected and the valence of each vertex is constant. So we need to show that there are no loops.

Let X = {a_{1}, …, a_{s}, b_{1}, …, b_{s}} be the set of generators, then if there is a loop in the Cayley tree, γ = v_{0}, …, v_{n}, such that v_{0} = v_{n}, then there must be a reduced word W on X such that v_{0}W = v_{n} = v_{0}. Thus W = [e]. W cannot be a_{i}a_{i}^{−1} or b_{j}^{2} = b_{j}b_{j}^{−1}, since both these imply that two consecutive edges in the loop are inverses of each other. Thus W = e.

Thus there are no loops, so the Cayley graph is a homogeneous tree.

Finally, there are 2(t + s) possible directed edges out of e, corresponding to each element of X ∪ X^{−1}, but since b_{i} = b_{i}^{−1}, s of these have been double-counted. Thus there are 2t + s edges from e, so q + 1 = 2t + s.

It turns out that these are the only groups which have locally finite Cayley graphs which are trees (see section 1.8).

We now discuss a generalisation of the free product, the amalgamated product.

Definition 1.2.7 Given a family of (disjoint) groups, {G_{i}}_{i∈I} for some index set I, a group A, and a collection of injective homomorphisms ϕ_{i} : A → G_{i}, for all i ∈ I (that is we can think of A as a subgroup of G_{i} for all i). Then we define the amalgamated product of the G_{i} to be

∗_{A} G_{i} = \frac{∗_{i∈I} G_{i}}{N}

where N is the normal closure of the set

{ϕ_{i}(a)ϕ_{j}(a)^{−1} : i, j ∈ I, a ∈ A}

in ∗_{i∈I} G_{i}.

Intuitively, this is “amalgamating” all the “copies” of A in the group ∗_{i∈I} G_{i} into just

one “copy”.

More concretely, it can be shown (Serre [11, §1.2, p2]), that the elements of ∗_{A} G_{i} can

be represented as pairs (a, s_{1} … s_{n}), where a ∈ A and s_{i} ∈ S_{j} \ {1}, S_{j} a set of right coset representatives of ϕ_{j}(A) in G_{j}, and if s_{i} ∈ S_{j} \ {1} then s_{i+1} ∉ S_{j} \ {1}.

To conclude this section we look at a generalisation of the direct product, the semidirect product.

Definition 1.2.8 Given a group G, N ⊲G, and H ≤ G such that G = HN and H ∩N = 1, then G is said to be the internal semidirect product of N and H, written G = H⊳< N or G = N >⊲H.

From Robinson [10, §1.5, p27], we get that each element of G has a unique expression of the form hn (h ∈ H, n ∈ N) and that conjugation of N by an element of H gives an automorphism of N, so we get a homomorphism α : H → Aut(N) (in other words H acts on N). We note that if this homomorphism is the trivial homomorphism, then G = H × N.

These facts allow us to reverse the definition and construct a G from H and N.

Definition 1.2.9 Given two groups H and N and a homomorphism α : H → Aut(N), we define the (external) semidirect product of H and N, written H⊳<_{α} N or N >⊲_{α}H, as the set {(h, n) : h ∈ H, n ∈ N} with multiplication defined by

(h_{1}, n_{1})(h_{2}, n_{2}) = (h_{1}h_{2}, α(h_{2})(n_{1})n_{2})

We notice again that if α is the trivial homomorphism, then the semidirect product and the direct product coincide. The definition of the multiplication is motivated by the fact that we are trying to simulate conjugation by h_{2} and the fact that (h_{1}n_{1})(h_{2}n_{2}) = h_{1}h_{2}(h_{2}^{−1}n_{1}h_{2})n_{2} always.

Robinson shows that if G = H⊳<_{α} N then G is also the internal semidirect product of H and N when imbedded in G by h ↦ (h, e) and n ↦ (e, n) respectively. Thus we will, in future, simply refer to the semidirect product of two groups.

1.3 Metrics on Trees

One can define a metric d_{V} on the vertices of a tree (V, E) in the following way:

Definition 1.3.1 The distance d_{V} (v, u) between two vertices v, u ∈ V is the length of the unique chain between v and u, or 0 if v = u.

Proof:

d_{V} is by definition a map from V to {0, 1, 2, …}, so it certainly has the correct range for a metric.

By definition, d_{V} (v, v) = 0.

If d_{V} (v, u) = 0 then since all chains have a length of at least 1, there cannot be a chain between v and u, so they must be the same vertex.

If d_{V} (v, u) = n, then there is a chain v, v_{1}, …, v_{n−1}, u, but since (v_{i}, v_{i+1}) ∈ E implies (v_{i+1}, v_{i}) ∈ E, u, v_{n−1}, …, v_{1}, v is a chain, and thus it must be the unique chain between u and v. u, v_{n−1}, …, v_{1}, v has length n, so d_{V} (u, v) = d_{V} (u, v).

If d_{V} (v, u) = n, and d_{V} (u, w) = m then

d_{V} (v, u) + d_{V} (u, w) = length of v, v_{1}, …, v_{n−1}, u + length of u, u_{1}, …, u_{m−1}, w
= length of v, v_{1}, …, v_{n−1}, u, u_{1}, …, u_{m−1}, w
≥ length of v, v_{1}, …, v_{n−a}, u_{a}, …, u_{m−1}, w
≥ d_{V} (v, w)

So d_{V} is a metric on V.

The metric d_{V} on V gives sufficient information to construct the whole tree, since E is exactly the set of all pairs {v, u} where d_{V} (v, u) = 1.

Definition 1.3.2 Given some vertex v ∈ V, we define the ball of radius r at v by

B_{v}(r) = {u ∈ V : d_{V} (v, u) ≤ r}

and the boundary of B_{v}(r) as

∂B_{v}(r) = {u ∈ V : d_{V} (v, u) = r}

More generally, given a finite subtree T = (U, D), we define the boundary of T by

∂T = {u ∈ U : val_{T} (u) < val_{Γ}(u)}

Finally, for a subtree T, we define the diameter of T to be

diam(T) = max{d_{V} (u, v) : u, v ∈ T }

We will sometimes abuse notation by referring to the subtree (B_{v}(r), D), where D is the set of edges with both extremities in B_{v}(r) as B_{v}(r). In this case the two definitions of the boundary coincide. We will extend the definition of boundaries to infinite trees in section 1.4.

The metric topology on V, (V, d_{V}) is the discrete topology on V, and it is well known that we can construct a metric d_{C} on the compact sets of this topology in the following manner:

Definition 1.3.3 Given two compact (ie. finite, since (d_{V} , V) discrete) sets A, B ⊆ V, define

A_{r} = {v ∈ V : d_{V} (v, u) ≤ r, for some u ∈ A}

and B_{r} similarly, then

d_{C}(A, B) = inf{r ∈ Z : A ⊆ B_{r}andB ⊆ A_{r}}

In particular we note that since finite subtrees and edges can be represented as finite subsets of V, we can define metrics on the set of subtrees and on the set of edges as follows:

Definition 1.3.4 Given two finite subtrees T_{1} = (U_{1}, D_{1}) and T_{2} = (U_{2}, D_{2}) of a tree, then the distance between them is defined to be d_{T} (T_{1}, T_{2}) = d_{C}(U_{1}, U_{2}).

Given two edges y_{1} = {u_{1}, v_{1}} and y_{2} = {u_{2}, v_{2}} of a tree, we define the distance between the edges to be d_{E}(y_{1}, y_{2}) = d_{C}({u_{1}, v_{1}}, {u_{2}, v_{2}}).

In effect these two definitions are simply the restriction of the compact metric to the set of finite trees and the set of edges respectively. In all three cases the metric topology will again be the discrete topology on the appropriate set.

1.4 The Boundary of an Infinite Tree

From Section 1.1, we note that once the order reaches a certain size, (specifically when all arguments of the order are ≥ 2), both the homogeneous and semi-homogeneous trees become infinite. This leads to the possibility of having chains of infinite length and additional structure derived from them. However, the following discussion is not limited to homogeneous trees, but in fact holds for any infinite tree.

Definition 1.4.1 An infinite chain is an infinite sequence v_{0}, v_{1}, v_{2}, … of vertices such that for all i, {v_{i}, v_{i+1}} is an edge and v_{i} ≠ v_{i+2}.

Two chains v_{0}, v_{1}, v_{2}, … and y_{0}, y_{1}, y_{2}, … are equivalent if there is some n and some m for which v_{n+k} = u_{m+k}, ∀ k ≥ 0.

The boundary, Ω, of an infinite tree is the set of equivalence classes of infinite chains.

If the infinite chain v_{0}, v_{1}, v_{2}, … is in the equivalence class ω ∈ Ω then we say ω is an endpoint of v_{0}, v_{1}, v_{2}, . . ..

Sometimes it will be convenient to think of the boundary of a tree as being the set of all infinite chains of the form o, v_{0}, v_{1}, v_{2}, … where o is some fixed vertex. These two concepts are equivalent, since the chains o, v_{0}, v_{1}, v_{2}, … are merely a complete set of representatives from each equivalence class of Ω.

Definition 1.4.2 A doubly infinite chain is a two-sided infinite sequence of vertices, …, v_{−2}, v_{−1}, v_{0}, v_{1}, v_{2}, … such that for all i, {v_{i}, v_{i+1}} is an edge and v_{i} ≠ v_{i+2}.

We now want to extend the definition of geodesics to include the boundary.

Definition 1.4.3 We call the infinite chain v_{0}, v_{1}, v_{2}, … ∈ ω the (infinite) geodesic from v_{0} to ω, written [v, ω), and the doubly infinite chain …, v_{−2}, v_{−1}, v_{0}, v_{1}, v_{2}, … with v_{0}, v_{1}, v_{2}, … ∈ ω_{+} and v_{0}, v_{−1}, … ∈ ω_{−} the (infinite) geodesic from ω_{−} to ω_{+}, written (ω_{−}, ω_{+}).

For any given v ∈ V, ω, ω_{+}, ω_{−} ∈ Ω, the infinite geodesics [x, ω) and (ω_{−}, ω_{+}) are unique.

We can now define a topology on V ∪ Ω as follows.

Definition 1.4.4 We define the topology by setting each element of V open, and define a basis of open neighbourhoods for each element of Ω.

Let v ∈ V, ω ∈ Ω. The for each u ∈ [v, ω) we define the neighbourhood N_{ω}(v, u) of ω to be the set of all vertices and end points of the infinite chains [x, ν) such that [v, u]∩[x, ν) = {u} (ie. the infinite chains which contain no vertex of [v, u] except u).

See Figure 1.4 for a diagram of what the open neighbourhoods look like. Note that there is considerable duplication amongst the neighbourhoods: If [v, ω) ∩ [u, ω) = [x, ω) then ∀ y ∈ [x, ω), N_{ω}(v, y) = N_{ω}(u, y), and if [v, u] ⊆ [v, ω) ∩ [v, ν) then N_{ω}(v, u) = N_{ν}(v, u).

Theorem 1.4.1 V ∪ Ω is compact.

Figure 1.4: An Open Neighbourhood of ω 15 Proof:

Choose some vertex o ∈ V. Let a_{r} = max{val(v) : v ∈ ∂B_{o}(r)}. Let Λ = (U, D) be a tree for which there is some o ∈ U such that val(v) = a_{d_{U} (v,o)} for all v ∈ U. Clearly Γ is isomorphic to a subtree of Λ. Let Δ be the boundary of Λ. We will show U ∪ Δ is compact.

Firstly we can identify Δ with Π = Z_{a_{0}}×∏_{i=1}^{∞} Z_{a_{i}−1} since the chain from o to δ ∈ Δ has a_{0} choices for the second element, and a_{i} − 1 for the (i + 1)st element of the chain. Furthermore this map is a homeomorphism since the product topology on Π has a base of neighbourhoods where given γ ∈ Π, n ∈ Z, we have the neighbourhood N_{γ}(n) = {λ ∈ Π : λ_{i} = γ_{i}, 0 ≤ i ≤ n}, which exactly the corresponds with restriction of the open neighbourhoods of U ∪ Δ to Δ. Thus Δ is compact, since by Tychanoff’s theorem, Π is compact.

Let {W_{i}}, i ∈ I be an open cover of basic sets of U ∪ Δ. Then {W_{i} ∩ Δ}, i ∈ I is an open cover of Δ. Therefore there is a finite subcover {W_{j} ∩ Δ}_{j=1}^{m} of Δ.

Now there must be some m such that

∂B_{o}(r) ⊆ ⋃_{j=1}^{l} W_{j}

otherwise we would be able to construct an element of Δ which was not covered by {W_{j} ∩ Δ}_{j=1}^{m}.

So (U ∪ Δ) \ (⋃_{j=1} lW_{j}) ⊆ B_{o}(r). Now for each v ∈ B_{o}(r), there exists some W_{k}, k ∈ I such that v ∈ U_{k}. Since |B_{o}(r)| is finite, we can append this finite number of additional sets to {W_{j}}_{j=1}^{m} and get a finite subcover of the whole set.

Thus U ∪ Δ is compact. This means that V ∪ Ω is compact, since we can get V ∪ Ω from U ∪ Δ by “pruning” Λ down to Γ. Each time a vertex is removed from Λ, an open neighbourhood is removed from U ∪ Δ. (See figure 1.5). Thus V ∪ Ω is closed, since it is homeomorphic to U ∪ Δ minus some union of open sets. So V ∪ Ω is compact.

Corollary 1.4.2 Ω is compact.

Proof:

V ∪Ω is compact. V is a union of vertices, which are all open, so V is open. Ω = (V ∪Ω)\V. Thus Ω is closed, so Ω is compact.

Proposition 1.4.3 V is dense in V ∪ Ω.

Proof:

Consider the geodesic [v_{0}, ω). The sequence {v_{i}}_{i=1}^{∞} clearly converges to ω. For each ω ∈ Ω there exists such a sequence, so we have our result.

We can also define a collection of measures on Ω.

Definition 1.4.5

Ω(u, v) = {ω ∈ Ω : [u, v] ⊂ [u, ω)}

Figure 1.5: “Pruning” Λ 17

Then {Ω(u, v) : v ∈ ∂B_{u}n} partitions Ω into |∂B_{u}n| open and compact sets. We can then use these partitions to generate a σ-algebra on Ω.

Definition 1.4.6 Given a path v_{0}, …, v_{n}, we define the following function

p_{v_{0}}(v_{n}) = val(v_{0}) × ∏_{i=1}^{n−1} val(v_{i})

\frac{1}{p_{v_{0}}(v_{n})} is the probability that a random walk starting at v_{0} will pass through v_{n}. We use this function to define a probability measure ν_{u} on the σ-algebra above, by setting ν_{u}(Ω(u, v)) = \frac{1}{p_{u}(v)}. Since this is a probability measure, ν_{u}(Ω) = 1.

Furthermore, since the Ω(u, v) are a basis for the relative topology on Ω, we can extend the σ-algebra to the Borel σ-algebra, and the measure to a Borel measure.

1.5 Automorphisms of Trees

Now, following Figà-Talamanca [4], we consider mappings of trees onto themselves.

Definition 1.5.1 An automorphism of a tree (V, E) is a bijective map g : V → V which preserves edges. ie. g(u) and g(v) are adjacent iff u and v are adjacent.

These automorphisms are exactly the surjective isometries of the metric space (V, d_{V}).

Lemma 1.5.1 g is an automorphism of (V, E) iff g is an isometry of (V, d_{V}).

Proof:

If g is an isometry of (V, d_{V}), then g(v) and g(u) are adjacent iff d_{V} (g(v), g(u)) = 1 iff d_{V} (v, u) = 1 (since g is an isometry) iff v and u are adjacent. So g is an automorphism of (V, E).

Conversely, if g is an automorphism of (V, E), then if d_{V} (v, u) = n, there exists a unique chain v, v_{1}, …, v_{n−1}, u of length n. But g(v), g(v_{1}), … g(v_{n−1}), u is a chain, as for all i, g(v_{i}) and g(v_{i+1}) are adjacent since v_{i} and v_{i+1} are adjacent, and g(v_{i}) ≠ g(v_{i+1}) since g is a bijection. g(v), g(v_{1}), … g(v_{n−1}), u has length n, so d_{V} (g(v), g(u)) = n. So g is an isometry of (V, d_{V}).

We can now split automorphisms of a tree into three classes.

Definition 1.5.2 An automorphism g of (V, E) is:

i. A rotation about o if g stabilises some vertex o, ie. ∃ o ∈ V such that g(o) = o.

ii. An inversion about an edge e if g stabilises e, but exchanges its endpoints. iii. A translation of step j along a geodesic γ if there is a geodesic

γ = …, v_{−2}, v_{−1}, v_{0}, v_{1}, v_{2}, …

such that g(v_{n}) = v_{n+j} for all n.

Theorem 1.5.2 For all automorphisms g of a tree (V, E), g is either a rotation, an inversion or a translation.

Proof:

Let v ∈ V be a vertex such that d_{V} (v, g(v)) = min{d_{V} (u, g(u)) : u ∈ V } = j.

If j = 0, then g(v) = v, so g is a rotation.

If j = 1, then if g^{2}(v) = v, then g interchanges v and another adjacent vertex, so g must be an inversion. If g^{2}(v) ≠ v then let γ be a geodesic,

γ = …, v_{−2}, v_{−1}, v_{0}, v_{1}, v_{2}, …,

where v_{n} = g^{n}(v). Then g(g^{n}(v)) = g^{n+1}(v), so g is a translation of step 1 along γ.

If j ≥ 2 then let [v, g(v)] = v, v_{1}, …, v_{j−1}, g(v). We note that g(v_{1}) ≠ v_{j−1}, since d_{V} (v_{1}, v_{j−1}) = j − 2 < j. We can now extend [v, g(v)] by letting v_{j+k} = g(v_{k}) for all k. v_{j} and v_{j+1} are adjacent for all n since g is an automorphism, and v_{j} ≠ v_{j+2} since g is bijective, so

γ = …, v_{−2}, v_{−1}, v_{0}, v_{1}, v_{2}, …, v_{j−1}, g(v), v_{j+1}, …

is a geodesic for which g(v_{n}) = v_{n+j}. So g is a translation of step j along γ.

So we have our result.

Note another result undelies the case j ≥ 2.

Lemma 1.5.3 Let g be an automorphism, and v a vertex. Let

v, v_{1}, …, v_{n−1}, g(v) = [v, g(v)]

be the chain joining v and g(v). If g(v_{1}) ≠ v_{n−1}, then g is a translation of step n along a doubly infinite chain of which [v, g(v)] is a subchain.

Proof:

This is proved by extending the chain [v, g(v)] exactly as for the case j ≥ 2 above. For a full proof, see Figà-Talamanca [4, I, §3, p10].

For a homogeneous tree of order ≥ 2 there are always all three types of automorphisms, however this is not the case for other types of trees. Any finite tree cannot have translations, since there are no infinite geodesics, thus the only possible automorphisms of a finite tree are rotations and inversions. Also, for any automorphism, g, the valence of v and the valence of g(v) must be the same, so in any tree for which there are no two adjacent vertices with the same valence, there are no inversions, and all translations must have step ≥ 2. So a

Figure 1.6: Diagram for Proposition

semi-homogeneous tree has only rotations and translations of even step, since translations of odd step would map vertices of different valencies onto each other.

It is also worthwhile noting that the type of automorphism is preserved under conjugacy, ie. gg′g^{−1} is of the same type as g′.

Also, clearly if g is a translation of step j along a geodesic γ, then g^{n} is a translation of step |nj| along γ.

Proposition 1.5.4 Given a tree, Γ = (V, E), we have:

i. The composition of two inversions on distinct edges is a translation of even step.

ii. The composition of an inversion about an edge and a rotation which does not fix both vertices of the edge is a translation of odd step on a geodesic containing that edge.

Proof:

i. Let h and g be inversions on the distinct edges x = {v, u} and y = {w, z}, ie. d_{E}(x, y) ≠ 0. See Figure 1.6.

If d_{E}(x, y) = 1, then the two edges have 1 point in common. Without loss of generality, let u = w, so gh(v) = g(u) = z and gh(u) = g(v) ≠ u, so gh is a translation of step 2, by Lemma 1.5.3.

If d_{E}(x, y) = n ≥ 2, then without loss of generality let [u, w] ⊂ [v, z], so d_{V} (u, w) = n − 1. Thus g(u) ∉ [u, z] and g(v) ∉ [u, g(u)], so u ∈ [v, gh(v)] but gh(u) ∉ [v, g(v)] so by Lemma 1.5.3, gh is a translation. Finally, d_{V} (v, gh(v)) = d_{V} (v, u) + d_{V} (u, w) + d_{V} (w, z) + d_{V} (z, g(a)) = 2n, so the translation is of even step.

ii. Let g be a rotation about v and h an inversion on {u, w}. See Figure 1.6.

If g(u) = u but g(w) ≠ w, then gh(w) = g(u) = u but gh(u) = g(w) ≠ w so gh is a translation of step 1.

If g(u) ≠ u and g(w) ≠ w, then let z be a point such that g(z) = z and that d_{V} (z, u) is minimum. Without loss of generality, we assume that d_{V} (z, w) = d_{V} (z, u) + 1. Then d_{V} (gh(w), w) = d_{V} (gh(w), u) + 1 = 2d_{V} (z, u) + 1 and gh(u) = g(w) ∉ [gh(w), w] = [g(u), w], so again by Lemma 1.5.3, gh is a translation of odd step.

1.6 The Automorphism Group of a Tree

Having investigated the properties of the automorphisms, we now turn to look at the properties of the automorphism group of Γ = (V, E), which we denote by Aut(Γ). As we proved in Section 1.5, this group is the same as the group of isometries.

We can turn Aut(Γ) into a topological group via the compact open topology:

Proposition 1.6.1 Given g ∈ Aut(Γ), and F ⊆ V, with F finite, then let

U_{F} (g) = {h ∈ Aut(Γ) : g(x) = h(x), ∀ x ∈ F }

i. These U_{F}, together with Ø and Aut(Γ) form a basis for a topology τ on Aut(Γ).

ii. The group operations of Aut(Γ) are continuous with respect to τ.

Proof:

i. To show that this is a base, all we need show is that intersections of these sets are some union of elements of the base. Intersections with Ø and Aut(Γ) satisfy this condition trivially.

So given two finite subsets of V, F and H,

U_{F} (f) ∩ U_{H}(h) = {g ∈ Aut(Γ) : g(x) = f(x), ∀x ∈ F, and g(y) = h(y), ∀y ∈ H}

If U_{F} (f) ∩ U_{H}(h) = Ø, then we are done, so we can assume that there exists some r ∈ U_{F} (f) ∩ U_{H}(h). Then

U_{F} (f) ∩ U_{H}(h) = {g ∈ Aut(Γ) : g(x) = f(x) = r(x), ∀x ∈ F,
and g(y) = h(y) = r(y), ∀y ∈ H}
= {g ∈ Aut(Γ) : g(x) = r(x), ∀x ∈ F ∪ H}
= U_{F ∪H}(r)

ii. Let (g_{λ}, f_{λ}) be a net, λ ∈ (D, ⪰) which converges to (g, f) in Aut(Γ) × Aut(Γ). That is, given any neighbourhood U of (g, f), there exists some residual W ⊂ D such that (g_{λ}, f_{λ}) ∈ U for all λ ∈ W.

Now given any finite subset H of V, we consider the neighbourhood U_{f^{−1}H}(g) × U_{f^{−1}H}(f), so for all λ ∈ W, (g_{λ}, f_{λ}) ∈ U_{f^{−1}H}(g) × U_{f^{−1}H}(f). Therefore,

f_{λ}(u) = f(u)  ∀ u ∈ f^{−1}H
⇒ f_{λ}^{−1}(v) = f^{−1}(v) ∀ v ∈ H
⇒ g_{λ}(f_{λ}^{−1}(v)) = g_{λ}(f^{−1}(v)), ∀ v ∈ H
= g(f^{−1}(v)) ∀ v ∈ H

since f^{−1}(v) ∈ f^{−1}H.

Thus g_{λ}f_{λ}^{−1} ∈ U_{H}(gf^{−1}), ∀λ ∈ W. Since U_{H}(gf^{−1}) is a neighbourhood base about gf^{−1}, we have that g_{λ}f_{λ}^{−1} converges to gf^{−1}.

This means that (g, f) ↦ gf^{−1} is continuous, which gives the continuity of the group operations.

It is worthwhile noting here that the U_{F} are subgroups of Aut(Γ). This topology has several properties, the most important of which is that it is locally

compact.

Definition 1.6.1 A topological group is locally compact if there exists a compact base at the identity.

Before we can prove local compactness we need the following result:

Theorem 1.6.2 For all v ∈ V, the stabiliser of v in Aut(Γ),

K_{v} = U_{{v}}(e) = {g ∈ Aut(Γ) : g(v) = v}

is compact.

Proof:

Every g ∈ K_{v} gives a permutation on the set ∂B_{v}(n), for all n ≥ 1, since g is an isometry. So K_{v} acts as a subgroup of the symmetric group on B_{v}(n), S(r_{n}) where r_{n} = |B_{v}(n)|,

so we have a homomorphism α_{n} : K_{v} → S(r_{n}).

Now we consider the set ∏_{n} S(r_{n}). By Tychanoff’s theorem, it is compact, since it is a product of finite (and therefore compact) sets. We can map K_{v} into ∏_{n} S(r_{n}) by the map α : g ↦ (α_{1}(g), α_{2}(g), …). α is continuous, since π_{n} ∘ α = α_{n}, where π_{n} is the projection from ∏_{n} S(r_{n}) onto S(r_{n}), and the α_{n} are continuous. This follows from the fact that the topology on S(r_{n}) is the discrete topology, so we need only worry about the inverse image of some element s of S(r_{n}). If s ∉ the image of K_{v} then the inverse image is the empty set, so let g ∈ K_{v} be such that α_{n}(g) = s. Then α_{n}^{−1}(s) = U_{B_{v}}(n)(g) ∩ K_{v}, which is open.

Furthermore, α is clearly injective, since if two automorphisms map onto exactly the same element of ∏_{n} S(r_{n}), they have exactly the same action on Γ, and so they must be identical.

If g is in the complement of k, then for some n, and for some v_{1}, v_{2} ∈ ∂B_{v}(n) we have g(v_{1}) = v_{2} but the element there is some v_{3} ∈ [v, v_{1}] with d_{V} (v, v_{3}) = m such that g(v_{3}) is not the element of [v, v_{2}] of distance m from v. For a fixed n, the set of such g is clearly open, so the union over all n is also open. Thus the image of K_{v} is closed and therefore compact and hence K_{v} is itself compact.

Corollary 1.6.3 Aut(Γ) is locally compact.

Proof:

The sets U_{F} (e) form a neighbourhood base at e. But U_{F} (e) = ⋂_{v∈F} K_{v} is compact, since K_{v} is compact for all v.

This means that Aut(Γ) has a Haar measure defined on it (See section 2.1), and so we may hope to get information about the group’s representations from this. We do this in Chapter 2.

Example 1.6.1 Let Γ be the homogeneous tree of order 2, which we will label as follows:

[figure: v_{−2}; v_{−1}; v_{0}; v_{1}; v_{2}]

Clearly K_{v_{0}} ≅ Z_{2}. There is only one possible inversion on an edge, and the set of translations is isomorphic to Z.

By Proposition 1.5.4(i), we can generate all translations of even step and all inversions via two inversions on adjacent edges. Thus we have that this subgroup N of Aut(Γ) is isomorphic to the group

< {a, b} | a^{2} = b^{2} = e > = Z_{2} ∗ Z_{2}

This subgroup is normal, since if g is an inversion or even step translation, then hgh^{−1} is of exactly the same type, ie. is still in this subgroup.

K_{v_{0}} ∩ N = e and since the product of a rotation and an inversion on a distinct edge, by Proposition 1.5.4(ii) is a translation of odd step, we get that K_{v_{0}}N = Aut(Γ). Thus Aut(Γ) is the internal semidirect product of K_{v_{0}} and N.

So, Aut(Γ) ≅ (Z_{2} ∗ Z_{2}) >⊲Z_{2}.

In general, the structure is not quite so nice, as we will see in Section 1.8.

1.7 Point Stabiliser Subgroups, K_{v}

We now investigate some of the properties of K_{v}.

Proposition 1.7.1 If K_{v} and K_{v} are a point stabiliser subgroup then

i. For all g ∈ Aut(Γ), gK_{v}g^{−1} = K_{u} where u = g(v).

ii. K_{v} is totally disconnected.

iii. If Γ is homogeneous of order q + 1, then if d_{V} (v, u) = n then K_{v} ∩ K_{u} has index p_{v}(u) = (q + 1)q^{n−1} in K_{v}.

Proof:

i. Let u = g(v). If k ∈ K_{v} then k(v) = v, so

gkg^{−1}(u) = gkg^{−1}g(v) = gk(v) = g(v) = u

so gkg^{−1} ∈ K_{u}. Therefore K_{u} ⊆ gK_{v}g^{−1}.

If k′ ∈ K_{u} then k′(u) = u, so

g^{−1}k′g(v) = g^{−1}k′gg^{−1}(u) = g^{−1}k′(u) = g^{−1}(u) = v

so k′ ∈ g^{−1}K_{v}g, and thus K_{u} = gK_{v}g^{−1}.

ii. Let g, h ∈ K_{v}, with g ≠ h. Thus there is some vertex u such that g(u) ≠ h(u). Now for any k ∉ U_{{v,u}}(g), U_{{v,u}}(g) ∩ U_{{v,u}}(f) = Ø, since if l(u) = k(u), l(u) ≠ g(u). Thus U_{{v,u}}(g) ∩ ⋃_{f∉U_{{v,u}}(g)} U_{{v,u}}(f) = Ø. Furthermore any l ∈ K_{v} is either in U_{{v,u}}(g) or ⋃_{f∉U_{{v,u}}(g)} U_{{v,u}}(f) so we have K_{v} as a disjoint union of open sets. Furthermore h ∈ ⋃_{f∉U_{{v,u}}(g)} U_{{v,u}}(f).

Thus given any two automorphisms, we can find two disjoint open sets which cover all of K_{v}. Thus K_{v} is totally disconnected.

Figure 1.7: Restrictions of K_{o}^{r}

iii. For each w ∈ ∂B_{v}(n), choose g_{w} ∈ K_{v} such that g_{w}(w) = u. Such a g_{w} exists because Γ is homogeneous. Then K_{v} = ⋃_{w∈∂B_{v}(n)} g_{w}(K_{v} ∩ K_{u}), and the cosets g_{w}(K_{v} ∩ K_{u}) are distinct. Since |∂B_{v}(n)| = (q + 1)q^{n+1} = p_{v}(u), we are done.

Part (iii) of the proof above can be generalised to any tree where the valence of any vertex u is only a function of d_{V} (v, u).

We now want to get a better hold on the structure of the point stabiliser subgroups. Let Γ be homogeneous of order q + 1. Choose some o ∈ V. Let K_{o}^{r} be the restriction of

K_{o} to B_{o}(r). We note that K_{o}^{1} ≅ S(q + 1). Then for r ≥ 2, as sets,

K_{o}^{r} = K_{o}^{r−1} × (\underbrace{S(q) × ⋯ × S(q)}_{t times})

where t = |∂B_{o}(r − 1)| = (q + 1)q^{r−2}, since any given element of K_{o}^{r} when restricted to B_{o}(r − 1) is an element of K_{o}^{r−1} and each of the S(q) describes the restriction of K_{o}^{r} when restricted to the sets W_{u} = {v ∈ ∂B_{o}(r) : d_{V} (v, u) = 1} where u ∈ ∂B_{o}(r−1). See Figure 1.7.

So we can clearly embed K_{o}^{r−1} in K_{o}^{r} via the map α : K_{o}^{r−1} → K_{o}^{r}, where α(k) = (k, (1, …, 1)), and we can embed N = S(q) × ⋯ × S(q) in K_{o}^{r} via the map β : N → K_{o}^{r}, where β(s_{1}, …, s_{t}) = (1, (s_{1}, …, s_{t})). Clearly N ∩K_{o}^{r−1} = e and we can clearly express every element of K_{o}^{r} as a product of an element of K_{o}^{r−1} and an element of N, by the previous paragraph.

Furthermore, N is normal, since we note that N is exactly the vertex stabiliser of B_{o}(r−1), and k(s_{1}, …, s_{t})k^{−1}(B_{o}(r − 1)) = kk^{−1}(B_{o}(r − 1)) = B_{o}(r − 1).

Hence K_{o}^{r−1} = K_{o}^{r−1}⊳< N.

Now given any element of g ∈ K_{o} we can describe it uniquely by its action for each r on B_{o}(r), so we can write g = (k_{1}, k_{2}, …), but we note that we must have that the restriction of k_{i+1} to B_{o}(i) must be the same as the restriction of g to B_{o}(i).

So K_{o} can be represented as the subset of ∏_{r=1}^{∞} K_{o}^{r} such that if g = (k_{1}, k_{2}, …) ∈ K_{o}, then k_{i+1}|_{B_{o}(i)} = k_{i}.

Note that this means that as sets,

K_{o} = S(q + 1) × S(q) × S(q) × ⋯

which can be used to give more concretely that K_{o} is compact.

Although the above discussion holds for homogeneous trees, it can be generalised to K_{T} = U_{T} (e) for any tree Γ, and for any finite subtree T. In this case the sets which are acted upon by K_{T}^{r} are T_{r} = {v ∈ V : ∃ u ∈ T where d_{V} (v, u) ≤ r} (see Section 1.3), K_{T}^{1} may be a product of symmetric groups, one for each orbit of K_{T} on T_{1}, and the permutation groups also become products of symmetric groups, one for each orbit of K_{T} on W_{u} = {v ∈ T_{r} \ T_{r−1} : d_{V} (v, u) = 1} where u ∈ T_{r−1} \ T_{r}. (Note that here we define T_{−1} = ∂T).

1.8 Faithful Transitive Subgroups

We now look at a class of discrete subgroups of Aut(Γ), where Γ is homogeneous of order q + 1, in an attempt to get a better hold on the structure of Aut(Γ).

Definition 1.8.1 We say that a subgroup F of Aut(Γ) is a faithful transitive subgroup if for every u, v ∈ V, there is a f ∈ F such that f(u) = v (transitivity), and if F ∩ K_{v} = e for all v ∈ V.

We note that if F is a faithful transitive subgroup of Aut(Γ), then if we choose some vertex o, then the map f ↦ fo is a bijection, ie. there is a direct correspondence between elements of F and elements of V. Also F is discrete in Aut(Γ), since if g, f ∈ F, go = fo, then g^{−1}fo = o, thus g^{−1}f = e, so g = f, which means U_{o}(f) ∩ F = f.

If Γ is not homogeneous, then it is impossible for a group to be transitive in this sense, but it is possible to generalise and get a meaningful definition (see Section 1.9).

We can actually get a characterisation of all faithful transitive subgroups of Aut(Γ), but we need the following lemma.

Lemma 1.8.1 Given a vertex o ∈ V and a set A ⊂ Aut(Γ) such that |A| = q + 1 and

i. A = A^{−1}

ii. A(o) = {a(o) : a ∈ A} = ∂B_{o}(1)

then

i. given a finite sequence a_{1}, …, a_{n} of elements of A, then d_{V} (o, a_{1} … a_{n}(o)) ≤ n.

ii. given any vertex v ∈ V, with d_{V} (v, o) = n, it is possible to find a unique finite sequence of n elements a_{1}, …, a_{n} of A such that a_{1} … a_{n}(o) = v, and for this sequence a_{i}a_{i+1} ≠ e, for i = 1, …, n − 1.

Proof:

i. We use induction on n.

If n = 1, the result is true by assumption.

If the result is true for n = k, then given a_{1}, …, a_{k+1} ∈ A, then

d_{V} (o, a_{1} … a_{k+1}(o)) = d_{V} (a^{−1}(o), a_{2} … a_{k+1}(o))

since a_{1} is an isometry. But

d_{V} (a^{−1}(o), a_{2} … a_{k+1}(o)) ≤ d_{V} (a^{−1}, o) + d_{V} (o, a_{2} … a_{k+1}(o))
≤ 1 + k

So the result is true for k + 1.

ii. We note first that since |A| = q + 1 and |∂B_{o}(1)| = q + 1, the second condition on A implies that if a, b ∈ A and a(o) = b(o), then a = b. Thus for each u ∈ ∂B_{o}(1), there is a unique a ∈ A such that a(o) = u.

Again, we use induction on n.

If n = 1, existance is true by assumption, and uniqueness follows from above.

If the result is true for n = k, then given v ∈ V with d_{V} (o, v) = k + 1, let [o, v] = v_{0}, …, v_{k+1} be the chain between o and v. Then there is a sequence a_{1}, …, a_{k} such that a_{1} … a_{k}(o) = v_{k}, with a_{i}a_{i+1} ≠ e, for i = 1, …, k − 1. Now d_{V} (o, (a_{1} … a_{k})^{−1}(v)) = d_{V} (v_{k}, v) = 1 since each a_{i} is an isometry. So there is a unique a ∈ A such that a(o) = (a_{1} … a_{k})^{−1}(v), therefore a_{1} … a_{k}a(o) = v, and furthermore, a_{k}a ≠ e, since then v = a_{1} … a_{k}a(o) = a_{1} … a_{k−1}(o), and by part (i), d_{V} (o, v) = d_{V} (o, a_{1} … a_{k−1}(o)) ≤ k − 1, which is a contradiction.

So the result is true for k + 1.

Sets of the form of A are easy to find, for instance, a set of inversions, one inverting each edge radiating from o will do. Alternatively, two inversions could be replaced by a step 1 translation and the corresponding step 1 translation in the other direction along the same geodesic.

The above lemma leads to the following classification theorem for faithful transitive subgroups.

Theorem 1.8.2 Let A be a subset of Aut(Γ) as above. Then A generates a faithful transitive subgroup F of Aut(Γ). Furthermore F is isomorphic to the free product of F_{s} and t copies of Z_{2}, and these are the only faithful transitive subgroups.

Proof:

Every element a of A must either be an inversion or a step one translation, by the following argument: If a is a rotation, then d_{V} (v, a(v)) must be even, so d_{V} (o, a(o)) ≠ 1. If a is a translation of step j ≥ 2, then by the proof of Theorem 1.5.2, the minimum distance between any two vertices is j, so d_{V} (o, a(o)) ≠ 1. Thus a must either be a step one translation or an inversion. It is an inversion iff a^{2} = e.

Let {b_{1}, …, b_{t}} be the subset of A of inversions, and let {b_{t+1}, ..., b_{t+s}, b_{t+1}^{−1}, …, b_{t+s}^{−1}} be the step one translations. So elements of the subgroup F of Aut(Γ) generated by A are all words consisting of elements of A where no two adjacent elements are inverses, so

F ≅ < A | b_{i}^{2} = e = b_{t+j}b_{t+j}^{−1}, ∀i = 1, …, t, j = 1, …, s >

so the b_{t+1}^{−1}, …, b_{t+s}^{−1} are clearly redundant, and we can instead write

F ≅ < {b_{1}, …, b_{t+s}} | b_{i}^{2} = e, ∀i = 1, …, t >

which is exactly the definition of the free product of the free group on s symbols with t copies of Z_{2}, ie.

F ≅ F_{s} ∗ \underbrace{Z_{2} ∗ ⋯ ∗ Z_{2}}_{t times}

(See Theorem 1.2.2).

F is clearly transitive by part (ii) of the previous Lemma, and F is faithful by the uniqueness condition from part (ii), since if fg(v) = v, then there is some h such that fgh(o) = h(o) by part (ii), and this implies that gh(o) = f^{−1}h(o), which by uniquness means g = f^{−1}, so there are no rotations other than e.

Finally, if F is a faithful transitive subgroup, then choose some vertex o ∈ V, and let A = {g ∈ F : g(o) ∈ ∂B_{o}(1)}, ie. A(o) = ∂B_{o}(n). Then since F is faithful, |A| = |∂B_{o}(1)| = q + 1. Furthermore, since d_{V} (o, g^{−1}(o)) = d_{V} (g(o), o) = 1, then g^{−1} ∈ A, so A = A^{−1}, and thus A fits the hypothesis. Therefore A generates a faithful transitive subgroup, and this must be exactly the original subgroup F.

Note that this result means that the faithful transitive subgroups of a homogeneous tree Γ are exactly the groups which have Γ as their Cayley graph (see Theorem 1.2.2).

We now hope to use this subgroup along with the point stabilisers, to get additional information about the structure of Aut(Γ). We might hope that since F ∩ K_{v} = e, and by Example 1.6.1 that we might be able to form the semidirect product of the two subgroups and get the whole automorphism group. Unfortunately, F is not, in general, normal. However, we do have the following.

Proposition 1.8.3 Aut(Γ) = K_{o}F = F K_{o} and furthermore if g = f_{1}k_{1} = k_{2}f_{2}, k_{1}, k_{2} ∈ K_{o}, f_{1}, f_{2} ∈ F, then k_{1}, f_{1}, k_{2}, f_{2} are unique.

Proof:

Let g ∈ Aut(Γ). Then there exists a unique f_{1} ∈ F such that f_{1}(o) = g(o), so there is a unique k_{1} ∈ K_{o} such that k_{1} = f_{1}^{−1}g, since f_{1}^{−1}g(o) = o, and hence g = f_{1}k_{1}.

Similarly there is some vertex v such that g(v) = o, so there is a unique f_{2} ∈ F such that f_{2}(v) = o. So there is a unique k_{2} = gf_{2}^{−1} ∈ K_{o}, since gf_{2}^{−1}(o) = g(v) = o, and hence g = k_{2}f_{2}

This means that, as sets, Aut(Γ) = F × K_{o} = K_{o} × F, which allows is to prove that the Haar measure on Aut(Γ) is unimodular (see section 2.1).

1.9 Extensions and Generalisations

In this section we deal with some related topics to those we have already discussed.

Unimodular Subgroups of Aut(Γ)

Although we have not yet defined unimodularity, now is an appropriate time to discuss what conditions might lead to unimodularity. If the reader is unfamiliar with the concept of unimodularity, then the appropriate definitions may be found in Section 2.1.

As we will see, if we want to perform harmonic analysis on a group which acts on a tree, it is critical that the above Proposition holds, since it is by this that we will prove unimodularity.

To retain the subgroup as being locally compact it has to be a closed subgroup, so only those which are closed need to be considered.

If we want unimodularity, by the method given here, then we must be able to write the subgroup G of Aut(Γ) as a product of a compact subgroup K of Aut(Γ) and a faithful transitive subgroup F.

However, this is not the only way of showing unimodularity. Figà-Talamanca [4] uses the idea of Gelfand pairs to show unimodularity. It turns out that it is sufficient to show that a group is both transitive on the vertices of Γ and on its boundary Ω to show that the group is unimodular.

Clearly these sorts of conditions fall far short of being neccessary, since any compact subgroup of Aut(Γ) is unimodular simply by virtue of being compact. In particular the point stabiliser subgroups K_{v}, and more generally K_{T} are unimodular, but they are clearly neither contain faithful transitive subgroups, nor are they transitive on vertices and even in the case of K_{T} neccesarily, they are not neccesarily even transitive on Ω.

The Explicit Structure of a Group Acting on Tree

It is possible to get hold of the structure of a general group acting on a tree. These results come from Serre [11].

We begin with the following definition.

Definition 1.9.1 Given a tree Γ = (V, E), and a subgroup G of Aut(Γ), then we define

i. The quotient graph Γ/G = (U, D), where U is the set of orbits of G in V and D is the set of orbits of G in E, and an element a of U is the endpoint of an element b in D if there is some vertex v in a such that it is the endpoint of some y in b.

ii. A tree of representatives of the quotient graph is a subtree T of Γ such that T and Γ/G are isomorphic.

Example 1.9.1 If Γ is the semihomogeneous tree of order (q + 1, r + 1), then V has two orbits under the action of Aut(Γ): the vertices of order q + 1, and the vertices of order r + 1, and E has one orbit. Thus the quotient graph is a single edge and its two endpoints.

Therefore any edge of Γ and its endpoints is a tree of representatives.

Note that there is no reason why a tree of representatives must exist, however if G contains no inversions, then Γ/G is a tree, and there must be a tree of representatives (see Serre [11, §3.1, p25]).

The fact the G must contain no inversions, as it turns out, is not a major problem, since if we take the barycentric subdivision of Γ, G has a well defined action on Γ′ which is inversion free. We map each g ∈ G to g′ ∈ Aut(Γ′), where g′(v) = g(v), for v ∈ V and g′(y) = g′({u, v}) = {g(u), g(v)}, for y = {u, v} ∈ E. The fact this is inverse free is clear since if g is an inversion on Γ, then g stabilises some edge y, and so g′ is a rotation about the corresponding vertex of Γ′. (In fact it can be shown, that if Γ is not the homogeneous tree of order 2, then Aut(Γ) ≅ Aut(Γ′)).

Serre shows that it is possible to derive the structure of many groups from their actions on a tree, however the simplest example is the case when the tree of representatives is an edge y = {u, v}, then G ≅ K_{u} ∗_{K_{y}} K_{v}, the amalgamated product of the point stabilisers of u and v amalgamated over the point stabiliser of y, that is the {g ∈ G : gv = v, gu = u}. There are more general results, however they are outside the scope of this thesis.

Chapter 2

Representations of Groups on a Tree

In this chapter we develop some initial theory concerning harmonic analysis and representations, before setting out to classify and identify the irreducible representations of Aut(Γ). After the initial sections, we essentially follow chapter III of Figà-Talamanca [4].

2.1 Haar Measures

Most of the following section, except that dealing specifically with groups acting on trees, can be found in any good book on Harmonic Analysis. In particular, the proofs of the general theorems can be found in Halmos [6].

We begin by producing a particularly nice measure on a topological group.

Theorem 2.1.1 (Haar, Von Neumann) Given a locally compact group G, with Borel sets B there exists a Borel measure μ on G, with the following properties:

i. If V ≠ Ø is open, then μ(V) > 0.

ii. There exists an open set V with μ(V) < ∞.

iii. If X ∈ B, then

μ(gX) = μ(X), ∀ g ∈ G.

(ie. μ is left invariant).

iv. μ is regular, that is μ(E) = inf{μ(V) : E ⊂ V, V open} and μ(E) = sup{μ(C) : C ⊂ E, C compact}

Such a measure is called a left Haar measure and is unique up to a multiplicative constant, that is if μ and ν are both left Haar measures, then for some c ∈ R_{+}, μ(X) = cν(X), ∀ X ∈ B

Similarly, there exists a right Haar measure on G.

(Existence is due to Haar, and uniqueness due to Von Neumann). Proof: See Halmos [6, XI; §58,60].

Example 2.1.1 The following are some examples of Haar measures.

i. On R, + the Lebesgue measure is both a left and right Haar measure.

ii. On Z, + the counting measure is both a left and right Haar measure.

iii. If G is compact, then μ(G) < ∞, so we can normalise the measure so that μ(G) = 1.

iv. Similarly, if G is discrete, then we can normalise the measure by setting μ({e}) = 1, making μ the counting measure on G.

v. Given two locally compact groups G_{1} and G_{2}, with Haar measures μ_{1} and μ_{2}, then we can define μ_{1} × μ_{2} on G_{1} × G_{2}. This measure is a Haar measure on G_{1} × G_{2}.

vi. If G has finite Haar measure and H is a closed normal subgroup, then G/H is a locally compact group and has Haar measure μ∘θ^{−1}, where θ : G → G/H is the quotient map.

vii. If H is a subgroup of G, such that μ(H) > 0, then the restriction of μ to H is a Haar measure on H.

A natural question arises: When is the Haar measure both left and right invariant? That is, when is it bi-invariant? If this occurs, we call the group unimodular.

Proposition 2.1.2 The following groups are unimodular:

i. All locally compact Abelian groups.

ii. All compact groups.

iii. All discrete groups.

Proof: See Rieter [9, III, §3.5-6, p62].

Proposition 2.1.3 Aut(Γ) is unimodular.

Proof:

Since each element of Aut(Γ) can be expressed uniquely as a product of an element of K_{v} and an element of F, a faithful transitive subgroup, Aut(Γ) ≅ K_{v} × F as topological spaces. Let μ be right-invariant Haar measure on Aut(Γ). Choose some v ∈ V, since K_{v} is compact μ(K_{v}) < ∞, so we shall normalise μ so that μ(K_{v}) = 1.

Then μ_{u} = μ restricted to K_{u} is a Haar measure on K_{u}, since K_{u} is an open subgroup of Aut(Γ). Since K_{u} ∩ K_{v} has index (q + 1)q^{n−1} in K_{v} where d_{V} (u, v) = n, μ(K_{v} ∩ K_{u}) = μ_{v}(K_{v} ∩ K_{u}) = \frac{1}{(q+1)q^{n−1}}

Furthermore, the measures on each of the K_{v} are essentially identical, since for E ⊆ K_{fv}, μ_{fv}(E) = μ_{v}(f^{−1}Ef), as follows. μ_{v}(f^{−1}.f) is certainly a measure on K_{fv}. For all k ∈ K_{fv},

μ_{v}(f^{−1}kEf) = μ_{v}(f^{−1}kff^{−1}Ef)
= μ_{v}(f^{−1}Ef)

since f^{−1}kf ∈ K_{v} and μ_{v} is a bi-invariant Haar measure. Thus μ_{v}(f^{−1}.f) is a Haar measure, so μ_{v}(f^{−1}Ef) = cμ_{fv}(E). But

μ_{v}(f^{−1}(K_{v} ∩ K_{fv})f) = μ_{v}(K_{f^{−1}v} ∩ K_{v})
= \frac{1}{(q + 1)q^{n−1}}
μ_{fv}(K_{v} ∩ K_{fv}) = μ(K_{v} ∩ K_{fv})
= μ_{v}(K_{v} ∩ K_{fv})
= \frac{1}{(q + 1)q^{n−1}}

Since if d_{V} (v, fv) = n, then d_{V} (v, f^{−1}v) = n also. Therefore c = 1, and we have our result.

Given E ∈ Aut(Γ), E Borel, then we define for each u ∈ V, f ∈ F, E_{f}(u) = (E ∩ K_{u}f)f^{−1} ⊆ K_{u}. Intuitively, this is “slicing” E into sections, one for each coset of K_{u}, and then pulling these back to K_{u}. Therefore E is the disjoint union of E_{f}(u)f as f varies over F, ie. E = ⋃_{f∈F} E_{f}(u)f. Now since F is countable, we have

μ(E) = ∑_{f∈F} μ(E_{f}f)
= ∑_{f∈F} μ(E_{f})
= ∑_{f∈F} μ_{u}(E_{f})

for any u ∈ V.

Now we are in a position to prove unimodularity. Let E ⊆ Aut(Γ), E Borel.

Let k ∈ K_{v}, then

μ(kE) = ∑_{f∈F} μ_{v}((kE)_{f}(v))
= ∑_{f∈F} μ_{v}((kE ∩ K_{v}f)f^{−1})
= ∑_{f∈F} μ_{v}(k(E ∩ k^{−1}K_{v}f)f^{−1})
= ∑_{f∈F} μ_{v}(k(E ∩ K_{v}f)f^{−1})

= ∑_{f∈F} μ_{v}((E ∩ K_{v}f)f^{−1})
= ∑_{f∈F} μ_{v}(E_{f}(v))
= μ(E)

Let h ∈ F, then

μ(hE) = ∑_{f∈F} μ_{v}((hE)_{f}(v))
= ∑_{f∈F} μ_{v}((hE ∩ K_{v}f)f^{−1})
= ∑_{f∈F} μ_{v}(h(E ∩ h^{−1}K_{v}f)f^{−1})
= ∑_{f∈F} μ_{h^{−1}v}(h^{−1}h(E ∩ h^{−1}K_{v}f)f^{−1}h)
= ∑_{f∈F} μ_{h^{−1}v}((E ∩ h^{−1}K_{v}hh^{−1}f)f^{−1}h)
= ∑_{f∈F} μ_{h^{−1}v}((E ∩ K_{h^{−1}v}h^{−1}f)f^{−1}h)
= ∑_{g∈F} μ_{h^{−1}v}((E ∩ K_{h^{−1}v}g)g^{−1})
= ∑_{g∈F} μ_{h^{−1}v}((E_{g}(h^{−1}v)
= μ(E)

And since any element of Aut(Γ) can be expressed as a product of elements from these two sets, we are done.

From this proof we know that μ(K_{v}) = μ(K_{u}) < ∞ for all u, v ∈ V, so we can normalise the Haar measure in such a way that μ(K_{v}) = 1, for all v ∈ V. We will assume that this normalisation has taken place.

2.2 Representations

In this section we explore some general ideas from representation theory.

Definition 2.2.1 Given a group G,

A representation π of G is a homomorphism from G to the group of linear operators on some Hilbert Space H_{π}, ie. ∀ g, h ∈ G we have

π(g)π(h) = π(gh)

π(g^{−1}) = π(g)^{−1}

In addition, we require that ∀ ξ, η ∈ H_{π}, ⟨π(g)ξ, η⟩ is continuous as a function of g.

A representation is unitary if π(g) is a unitary operator for all g ∈ G. (Recall that an operator U on H_{π} is unitary if U^{∗} = U^{−1}, where U^{∗} is the operator such that ⟨Uξ, η⟩ = ⟨ξ, U^{∗}η⟩, ∀ ξ, η ∈ H_{π}).

A representation is irreducible if there is no nontrivial closed subspace of H_{π} invariant under the action of π. ie. the closed span of {π(g)ξ : g ∈ G} is H_{π}, ∀ ξ ∈ H_{π}.

A representation is faithful if it is injective.

Two representations π and ϕ are (unitarily) equivalent if there exists a surjective linear isometry U : H_{π} → H_{ϕ} such that Uπ(g) = ϕ(g)U for all g ∈ G.

Given two representations π and ϕ,we define their direct sum π⊕ρ to be the representation from G → H_{π} ⊕ H_{ϕ} defined by (π ⊕ ρ)(g) = (π(g), ρ(g)).

For the rest of this section we will assume that G is unimodular and that all representations will unitary.

There are two representations which should be introduced immediately.

Definition 2.2.2 The left regular representation λ on a locally compact group G, with Haar measure μ, is a representation on H_{λ} = L^{2}(G, μ) defined by

λ(g)f(k) = f(g^{−1}k)

where g, k ∈ G, f ∈ L^{2}(G).

The right regular representation ρ is defined similarly, but with

ρ(g)f(k) = f(kg)

There is also a class of representations that we will need to know more about. This section comes from Dixmier [2].

Definition 2.2.3 A representation π is square integrable if ∃ξ ∈ H_{π}, with ξ ≠ 0, such that g ↦ ⟨π(g)ξ, ξ⟩ is in L^{2}(G).

Proposition 2.2.1 If π is an irreducible square integrable representation, then

i. The coordinate functions t_{ξ,ν} : g ↦ ⟨π(g)ξ, ν⟩ are L^{2} for all ξ, ν ∈ H_{π}.

ii. There exists a number d_{π}, 0 < d_{π} < ∞, such that

∫_{G} t_{ξ,ν}(g)\overline{t_{ξ′,ν′}(g)} dμ(g) = \frac{⟨ξ, ξ′⟩⟨ν, ν′⟩}{d_{π}}

We call d_{π} the formal dimension of π.

Proof: See Dixmier [2, p278–282].

Now if we fix ν_{0}, where ∥ν_{0}∥^{2} = 1 and set (Uξ)(g) = \sqrt{d_{π}}t_{ξ,ν_{0}}(g), then from the above propsition,

i. Uξ ∈ L^{2}

ii. U is an isometry from H_{π} to L^{2}, since

⟨Uξ, Uξ′⟩ = d_{π} \frac{⟨ξ, ξ′⟩⟨ν_{0}, ν_{0}⟩}{d_{π}}
= ⟨ξ, ξ′⟩

iii. U intertwines π and ρ as follows:

(Uπ(g)ξ)(h) = \sqrt{d_{π}}⟨π(h)π(g)ξ, ν_{0}⟩
= (Uξ)(hg)
= ρ(g)(Uξ)(h)

This means that π is unitarily equivalent to some subrepresentation of ρ.

It is also possible to define a function s_{ξ,ν} : g ↦ ⟨ξ, π(g)ν⟩, in which case corresponding results give that π and λ can be intertwined, and that π is also unitarily equivalent to a subrepresentation of λ.

This means that given any irreducible square integrable representation π on H_{π} we can consider H_{π} as a subspace of L^{2} and π as the left (or right) regular representation restricted to this subset.

We also need the concept of induced representations. The following comes from Figà-Talamanca [4, III, §3, p132]. It concerns only the special case of induced representations of unimodular, separable, locally compact groups and compact open subgroups.

Let G be a unimodular, separable, locally compact group, and K a compact open subgroup of G. Let σ be a unitary representation on K.

Definition 2.2.4 Let S^{σ} be the space of functions f : G → H_{σ} such that

i. f(gh) = σ(h^{−1})f(g) for ever g ∈ G and h ∈ K.

ii. ∫_{G} ∥f(g)∥^{2} dg < ∞

This gives us that

∫_{G} ∥f(g)∥^{2} dg = μ(K) ∑_{x∈G/K} ∥f(g)∥^{2}

and S^{σ} is a Hilbert space with inner product

⟨f, g⟩ = ∫_{G}⟨f(x), g(x)⟩ dx

S^{σ} is invariant under left translation, so we can define a left regular representation of G on S^{σ}, which we denote Ind(σ), that is Definition 2.2.5 We define the induced representation of σ to be

Ind(σ)(x)f(t) = f(x^{−1}t)

Two representations σ_{1} and σ_{2} are unitarily equivalent, iff their induced representations are also unitarily equivalent.

Also given two representations σ_{1} and σ_{2}, we have that Ind(σ_{1}⊕σ_{2}) is unitarily equivalent to Ind(σ_{1}) ⊕ Ind(σ_{2}).

The if σ is square integrable, so is Indσ, and the formal dimension of Ind(σ) is dim(σ)/μ(K). If S_{K}^{σ} is the subspace of S^{σ} of functions supported on K, then S_{K}^{σ} is a closed non-trivial

subspace of S^{σ}. If we define for ξ ∈ H_{σ} the functions f_{ξ} where f_{ξ}(x) = σ(x^{−1})ξ for x ∈ K and f_{ξ}(x) = 0 otherwise, then the map ξ ↦ f_{ξ} is an isomorphism of H_{σ} onto S_{K}^{σ}.

Thus any function in S^{σ} with compact support is a linear combination of left translates of functions in S_{K}^{σ}, and the subspace of S^{σ} generated by ⋃_{g∈G} λ(g)S_{K}^{σ} is dense in S^{σ}.

Proposition 2.2.2 If σ is a unitary irreducible representation of K, then Ind(σ) is irreducible iff every closed non-trivial invariant subspace of S^{σ} contains a non-trivial function of S_{K}^{σ}.

Proof: See Theorem 3.13 of Figà-Talamanca [4, III, §3, p134].

2.3 Representations of Aut(Γ)

We now seek to classify all irreducible representations of Aut(Γ), and develop some tools to help deal with them.

If K is a compact subgroup of Aut(Γ) with normalised Haar measure μ_{K}, then given a unitary representation π on H_{π}, we fix some ξ ∈ H_{π}, and define a map

η ↦ ∫_{K}⟨π(k)ξ, η⟩ dμ_{K}(k)

This map is bounded conjugate linear map, so by the Riesz representation theorem, there exists some ξ′ ∈ H_{π} such that this map is

ν ↦ ⟨ξ′, ν⟩

ξ′ depends on ξ in a bounded linear fashion, and hence ξ′ is of the form P_{π}(K)ξ for some linear operator P_{π}(K). Furthermore, P_{π}(K) is bounded, since

|⟨P_{π}(K)ξ, η⟩| ≤ ∫_{K} |⟨π(K)ξ, η⟩| dμ_{K}(k)
≤ ∥ξ∥.∥η∥

So ∥P_{π}(K)∥ ≤ 1, and clearly now

⟨P_{π}(K)ξ, η⟩ = ∫_{K}⟨π(K)ξ, η⟩ dμ_{K}(k)

This leads to the following proposition: Proposition 2.3.1 For such a P_{π}(K) we have:

i. Given k ∈ K, we have π(k)P_{π}(K) = P_{π}(K). ii. If π(k)ξ = ξ, ∀ k ∈ K, then P_{π}(K)ξ = ξ. iii. P_{π}(K) is the orthogonal projection onto the space of π(k)-invariant vectors. Proof:

i. For all ξ, η ∈ H_{π},

⟨π(g)P_{π}(K)ξ, η⟩ = ⟨P_{π}(K)ξ, π(g)^{∗}η⟩
= ∫_{K}⟨π(k)ξ, π(g)^{∗}η⟩dk
= ∫_{K}⟨π(g)π(k)ξ, η⟩dk
= ∫_{K}⟨π(gk)ξ, η⟩dk
= ∫_{K}⟨π(k′)ξ, η⟩dk′
= ⟨P_{π}(K)ξ, η⟩

So π(k)P_{π}(K) = P_{π}(K). ii. For all η ∈ H_{π},

⟨P_{π}(K)ξ, η⟩ = ∫_{K}⟨π(k)ξ, η⟩dk
= ∫_{K}⟨ξ, η⟩dk
= ⟨ξ, η⟩ ∫_{K} dk
= ⟨ξ, η⟩

So P_{π}(K)ξ = ξ. iii. P_{π}(K) is an orthogonal projection if P_{π}(K)P_{π}(K) = P_{π}(K) and P_{π}(K)^{∗} = P_{π}(K).

This follows from (i) and (ii), since

⟨P_{π}(K)P_{π}(K)ξ, η⟩ = ∫_{K}⟨π(k)P_{π}(K)ξ, η⟩dk
= ∫_{K}⟨P_{π}(K)ξ, η⟩dk
= ⟨P_{π}(K)ξ, η⟩

and

⟨P_{π}(K)^{∗}ξ, η⟩ = ⟨ξ, P_{π}(K)η⟩
= ∫_{K}⟨ξ, π(k)η⟩dk
= ∫_{K}⟨π(k)^{∗}ξ, η⟩dk
= ∫_{K}⟨π(k)^{−1}ξ, η⟩dk
= ∫_{K}⟨π(k^{−1})ξ, η⟩dk
= ⟨P_{π}(K)ξ, η⟩

as required.

If T is a finite subtree of Γ, then let K_{T} be the subgroup of G that fixes the points of T, and let P_{π}(T) = P_{π}(K_{T}). P is well-defined, since K_{T} is the intersection of the stabilisers of each of the vertices of T, and is thus the intersection of a finite number of compact subgroups, and is so therefore itself compact. Similarly, since each of the point stabiliser subgroups is open, K_{T} is open. In particular μ_{G}(K_{T}) > 0, so by the properties of Haar measures,

P_{π}(T)ξ = \frac{1}{μ_{G}(K_{T})} ∫_{K_{T}} π(k)ξ dμ_{G}(k)

Also, as T varies over all finite complete subtrees, the K_{T} form an open neighbourhood basis of e. This gives us that for all ξ ∈ H_{π}, ξ ≠ 0, there is a finite complete subtree T such that

∫_{K_{T}} ⟨π(k)ξ, ξ⟩ dμ_{K_{T}} (k) ≠ 0

This follows since ⟨π(g)ξ, ξ⟩ is continuous in g, and

⟨π(e)ξ, ξ⟩ = ⟨ξ, ξ⟩ = ∥ξ∥^{2} ≠ 0

giving a real number ε such that 0 < ε < ∥ξ∥^{2}, so there is a finite complete subtree T for which π(K_{T})ξ ⊆ {η ∈ C : |η − ∥ξ∥^{2}| < ε}, so Re(π(g)ξ) > 0 for all g ∈ K_{T}, so

Re(∫_{K_{T}} ⟨π(k)ξ, ξ⟩ dμ_{K_{T}} (k)) > 0

giving the result.

A direct consequence of this is that for each ξ ≠ 0, there is a tree T such that P_{π}(T)ξ ≠ 0. Thus we can make the following definitions:

Definition 2.3.1 Let π be an irreducible unitary representation of G. Then there is some largest positive integer ℓ_{π} such that if T is a finite complete subtree for which P_{π}(T) ≠ 0 then ℓ_{π} ≤ the number of vertices of T.

If ℓ_{π} = 1 then π is called spherical.

If ℓ_{π} = 2 then π is called special.

If ℓ_{π} > 2 then π is called cuspidal.

A finite complete subtree T with ℓ_{π} vertices, for which P_{π}(T) ≠ 0 is called a minimal tree of π.

We note that if T is a minimal tree, then gT is also a minimal tree for all g ∈ G, since P_{π}(gT) = π(g)P_{π}(T)π(g)^{−1} ≠ 0, and the number of vertices is clearly the same.

Definition 2.3.2 If T is any complete finite subtree, then we define

i. H_{π}(T) to be the space of all K_{T} -invariant vectors, ie. H_{π}(T) is the range of P_{π}(T).

ii. V_{π} to be the union over all finite complete subtrees T of the H_{π}(T).

It is easy to show (Figà-Talamanca [4, III, §1, p106]) that V_{π} is dense in H_{π}, even if π is not irreducible.

Let π be a unitary representation, and let T be a subtree of π with ℓ_{π} vertices. Thus if T^{∘} is a proper complete subtree of T, then P_{π}(T^{∘})ξ = 0, for every ξ ∈ H_{π}. Thus if ξ, η ∈ V_{π} and P_{π}ξ = ξ, then the coefficient function u(g) = ⟨π(g)ξ, η⟩ has the following properties:

i. u(gk) = ⟨π(g)π(k)ξ, η⟩ for all k ∈ K_{T}.

ii. There is a finite complete subtree T′ such that u(kg) = ⟨π(g)ξ, π(k^{−1})η⟩ for all k ∈ K_{T′}. (T′ is any subtree for which P_{π}(T′)η = η).

iii. If T^{∘} is a proper complete subtree of T, then

\frac{1}{μ_{G}(K_{T^{∘}})} ∫_{K_{T^{∘}}} u(gk) dμ_{G}(k) = ⟨P_{π}(T^{∘})ξ, π(g^{−1})η⟩ = 0

for all g ∈ G.

To study the coefficient functions more closely, we generalise the notion as follows:

Definition 2.3.3 Let T be a complete finite subtree of Γ. We let S(T) be the linear space of continuous functions u satisfying the following properties:

i. u is right-K_{T} -invariant.

ii. u is left-K_{T_{u}}-invariant for some finite complete subtree T_{u}, dependant on u.

iii. for every proper complete subtree T^{∘} ⊂ T

∫_{K_{T^{∘}}} u(gk) dμ_{G}(k) = 0

for all g ∈ G.

S(T) is G-left-invariant since, if u ∈ S(T), then for any g ∈ G, v(h) = u(gh) ∈ S(T), as

i. For all k ∈ K_{T}, we have v(hk) = u(ghk) = u(gh) = v(h), since u is right-K_{T} -invariant.

ii. Given k ∈ K_{g^{−1}T′_{u}}, we have v(kh) = u(gkh) = u(gg^{−1}fgh) = u(fgh) = u(gh) = v(h), where f ∈ K_{T′_{u}}. Thus v is left-K_{g^{−1}T′_{u}}-invariant.

iii. Given T^{∘} ⊂ T, h ∈ Aut(Γ),

∫_{K_{T^{∘}}} v(hk) dμ_{G}(k) = ∫_{K_{T^{∘}}} v(hk) dμ_{G}(k)
= ∫_{K_{T^{∘}}} u(ghk) dμ_{G}(k)
= 0

Thus if we could show that S(T) ⊆ L^{2}(Aut(Γ)), S(T) would be an invariant subspace for λ.

It is easy to see, using similar arguments to above, that under right multiplication by g, S(T) becomes S(gT).

Now for a spherical representation π, if we consider S(v), where v is the minimal tree for π, then condition (iii) becomes trivial, so it is difficult to use S(T) to extract any information about π.

2.4 Special Representations

Let π be a special representation. Therefore all minimal trees of π must be edges. Let y = {a, b} ∈ E be an edge which is minimal for π.

Proposition 2.4.1 If π is a special representation, then all edges y′ = {a′, b′} ∈ E are minimal.

Proof:

If y is a minimal edge, then there is a g ∈ Aut(Γ) such that y′ = gy. Thus P_{π}(y′) = π(g)P_{π}(y)π(g^{−1}) ≠ 0.

Consider S(y). We can replace, for some n_{u}, T_{u} by B_{y}(n_{u}) which is the tree with vertices {v ∈ V : d_{V} (v, w) ≤ n_{u} for some w ∈ y} since for n_{u} sufficiently large, T_{u} ⊆ B_{y}(n_{u}), so K_{B_{y}(n_{u})} ⊆ K_{T_{u}}, giving the result. We will call this property (1).

Also remembering that S(y) is left invariant, we have, for all u ∈ S(y) and for all h, g ∈ Aut(Γ),

∫_{K_{a}} u(h^{−1}kg) dk = 0

so, if we set g = hf, then

0 = ∫_{K_{a}} u(h^{−1}khf) dk
= ∫_{h^{−1}K_{a}h} u(kf) dk
= ∫_{K_{ha}} u(kf) dk

We will call this property (2).

Now consider the following orientation of Γ. We define E_{+} to be the set of directed edges such that a is a local source and b is a local sink, and the pattern repeats throughout Γ. This is best described by illustration: see Figure 2.1. We define E_{−} = \overline{E_{+}}.

Since Aut(Γ) acts transitively on directed edges, and K_{y} stabilises [a, b], then we can identify G/K_{y} with E_{+} ∪ E_{−} via the map α : g ↦ [ga, gb]. So we define S̃ to be the space of functions defined by ũ ∈ S̃ iff u(g) = ũ(α(g)).

Since any edge y′ ∈ E has two directed edges associated with it, we define u_{+}(y′) to be the value of ũ on the positively oriented component of y′, and u_{−}(y′)to be the value of ũ on the negatively oriented component of y′.

We can now transfer the definition of S(y) across to S̃, and get the following Lemma, however first we define N (a, x) = N_{ω}(a, x) where ω is any infinite chain which starts with [a, x].

Lemma 2.4.2 A function ũ on E_{+} ∪ E_{−} is in S̃ iff

i. There exists an n > 0 such that ∀ x such that d_{V} (a, x) = n, then u_{+}(y′) and u_{−}(y′) depend only on the distance of y′ from a, d_{C}({a}, y′), for y′ ∈ N (a, x).

ii. For all v ∈ V, and m ∈ N,

∑_{d_{C}({v},y′)=m} u_{+}(y′) = 0

Figure 2.1: The Orientation E_{+} 43 and

∑_{d_{C}({v},y′)=m} u_{+}(y′) = 0

Proof:

These two properties are essentially the translation of conditions (1) and (2) to S̃.

i. Property (1) gives us that u is left-K_{B_{y}(n_{u})}-invariant, and so ũ must be constant on the orbits of K_{B_{y}(n_{u})} on E_{+} ∪ E_{−}. However, the orbits of K_{B_{y}(n_{u})} are exactly the edges in N (a, x) of a given distance and orientation from x ∈ ∂B_{y}(n_{u}), so they must all have the same value. Thus the two conditions are equivalent.

ii. Property (2) gives, given some vertex v,

∫_{K_{v}} u(kf) dk = 0

which translated to E_{+} ∪ E_{−} gives that the sum of ũ within an orbit of K_{v} must be 0. The edges of a given distance and orientation from v are exactly these orbits. So again the two conditions are equivalent.

Thus the lemma follows.

From this we can see that each element of S̃ is determined by its values on some finite set of edges, since all the edges outside B_{y}(n_{u}) are determined by the values on ∂B_{y}(n_{u}).

More specifically, if y′ ∈ N (a, x) and u_{+}({x, t}) = c where [a, x] = a, …, t, x, and d_{C}(\{x\}, y′) = k, then u_{+}(y′) = (−1)^{k+1} \frac{c}{q^{k}}. A similarly result holds for u_{−}. This result can be proved inductively, by considering first the edges of distance 1 from x: the sum on the positively oriented edges is 0, and since the q edges in N (a, x) which have x as an endpoint all have the same value, u_{+} = −\frac{c}{q}. Repeating this process on all vertices of a given distance from x proves the result.

Furthermore, we can show the following using this result.

Theorem 2.4.3 If ũ ∈ S̃, then ũ ∈ ℓ^{2}(E_{+} ∪ E_{−}) and unless ũ = 0, ũ ∉ ℓ^{1}(E_{+} ∪ E_{−}). This means that u ∈ S(y) is in L^{2}(Aut(Γ)), but not in L^{1}(Aut(Γ)) unless u = 0.

Proof:

We first note that

∑_{y′} |u_{+}(y′)|^{2} = ∑_{y′∈B_{y}(n_{u})} |u_{+}(y′)|^{2} + ∑_{y′∉B_{y}(n_{u})} |u_{+}(y′)|^{2}

but ∑_{y′∈B_{y}(n_{u})} |u_{+}(y′)|^{2} is a sum over a finite number of edges, and is hence finite. Now

∑_{y′∉B_{y}(n_{u})} |u_{+}(y′)|^{2} = ∑_{x∈∂B_{y}(n_{u})} ∑_{k=1}^{∞} ∑_{y′∈N (a,x),d_{C}({x},y′)=k} |u_{+}(y′)|^{2}

= ∑_{x∈∂B_{y}(n_{u})} ∑_{k=1}^{∞} ∑_{y′∈N (a,x),d_{C}(\{x\},y′)=k} |\frac{c(x)}{q^{k}}|^{2}

= ∑_{x∈∂B_{y}(n_{u})} |c(x)|^{2} ∑_{k=1}^{∞} q^{k} \frac{1}{q^{2k}}

= ∑_{x∈∂B_{y}(n_{u})} |c(x)|^{2} ∑_{k=1}^{∞} \frac{1}{q^{k}}
= ∑_{x∈∂B_{y}(n_{u})} |c(x)|^{2} \frac{q}{q − 1}
< ∞

Thus

∑_{y′} |u_{+}(y′)|^{2} < ∞

a similar argument gives

∑_{y′} |u_{−}(y′)|^{2} < ∞

and thus combining the two we get that ũ is ℓ^{2}.

Assume that ũ ≠ 0. Then there must be some x ∈ ∂B_{y}(n_{u}) such that u_{+}({t, x}) ≠ 0 (or u_{−}({t, x}) ≠ 0; we will assume without loss of generality that it is u_{+}), since if it were, then ũ would have to be 0 everywhere. Thus the ℓ^{1} norm of ũ must be at least the sum over the values of u_{+} on the y′ ∈ N (a, x), ie.

∥ũ∥_{1}^{2} ≥ ∑_{k=1}^{∞} ∑_{y′∈N (a,x),d_{C}({x},y′)=k} |u_{+}(y′)|

≥ ∑_{k=1}^{∞} ∑_{y′∈N (a,x),d_{C}(\{x\},y′)=k} |\frac{c(x)}{q^{k}}|

≥ ∑_{k=1}^{∞} q^{k} \frac{|c(x)|}{q^{k}}

≥ ∑_{k=1}^{∞} |c(x)|
= ∞

So therefore ũ is ℓ^{1} iff ũ = 0.

Finally we note that u ∈ S(y) has L^{2} norm equal to some multiple of the ℓ^{2} norm of the corresponding ũ, and similarly for L^{1}. (This multiple, as it turns out is μ(K_{y}) = \frac{1}{q+1}μ(K_{a}) = \frac{1}{q+1}). Thus the results carry directly over into S(y).

Proposition 2.4.4 The subspace of S(y) of all left-K_{y}-invariant functions is a two dimensional subspace, and for every f in this subspace

∥f∥_{2}^{2} = \frac{1}{q − 1}(|f_{+}(y)|^{2} + |f_{−}(y)|^{2})

Proof:

Let M be our left-K_{y}-invariant subspace. If f ∈ M, then f̃ is constant on edges of distance n from y. We note from above that this means that f̃ is determined completely by the values of f_{+} and f_{−} on y.

More specifically, if f_{+}(y) = c, then f_{+}(y′) = c(\frac{−1}{q})^{d_{E}(y,y′)}, and if f_{−}(y) = d, then f_{+}(y′) = d(\frac{−1}{q})^{d_{E}(y,y′)}.

Let ṽ_{1} be the function which is 1 on the positively oriented component of y and 0 on E_{−}, and let ṽ_{2} be the function which is 0 on E_{+} and 1 on the negatively oriented component of y. Then for any f ∈ M, f = cv_{1} + dv_{2}. Thus {v_{1}, v_{2}} is a complete basis for M.

Hence M is two dimensional.

Finally, from the above theorem, we can easily show that

∑_{y′∈E} |f_{+}(y′)|^{2} = |f_{+}(y)|^{2} \frac{q + 1}{q − 1}

and

∑_{y′∈E} |f_{−}(y′)|^{2} = |f_{−}(y)|^{2} \frac{q + 1}{q − 1}

Thus ∥f∥_{2}^{2} = (|f_{+}(y)|^{2} + |f_{−}(y)|^{2})\frac{q+1}{q−1} \frac{1}{q+1} = \frac{1}{q−1}(|f_{+}(y)|^{2} + |f_{−}(y)|^{2}) as required.

Note that in the context of S(y), this is saying that if u is in this subspace, then u is determined by its value on the set of automorphisms which stabilise y as a set.

So we have that S(y) is indeed a subset of L^{2}(Aut(Γ)), and as it is at least 2-dimensional, it is non-trivial. Let M(y) be the closure of S(y) in L^{2}(Aut(Γ)), so M(y) is a closed, nontrivial, invariant subspace of λ, (the left regular representation), harking back to the fact that S(y) is left-invariant.

Proposition 2.4.5 M(y) can be identified with the subspace of ℓ^{2}(E_{+} ∪ E_{−}) such that for all v ∈ V, and m ∈ N,

∑_{d_{C}({v},y′)=m} u_{+}(y′) = 0

and

∑_{d_{C}({v},y′)=m} u_{+}(y′) = 0

(ie. property (ii) of S̃).

Proof:

It suffices to check that all functions with this property are in the closure of S.

Choose some f ∈ ℓ^{2} such that this holds. Then define the sequence {u_{n}} ⊂ S(y) by setting ũ = f̃ on B_{y}(n) and letting property (i) of S̃ hold for ũ with n_{u} = n.

By similar arguments to the above Theorem and Lemma, it is possible to show that

∥f − u_{n}∥_{2}^{2} ≤ μ(K_{y})(1 + (q − 1)^{−1/2})^{2}( ∑_{y′∉B_{y}(n)}(|f_{+}(y′)|^{2} + |f_{−}(y′)|^{2}))

which tends to 0 as n goes to infinity.

Thus f is the limit of the u_{n} and we are done.

Let λ_{y} be the restriction of λ to S(y). We note that since ρ(g)S(y) = S(gy), so therefore ρ(g)M(y) = M(gy). Thus λ_{y} and λ_{gy} are unitarily equivalent.

Note that the bi-K_{y}-invariant elements of M(y) are exactly the bi-K_{y}-invariant elements of S(y), since if a function is both left- and right-K_{y}-invariant, it must be in S(y), by definition. Thus the subspace is still two dimensional.

In fact this space is exactly P_{λ}(y)(M(y)). Indeed, we can exactly specify the action of P_{λ}(y) on M(y), since, for f ∈ M(y), we have

(P_{λ}(y)f)(e) = ∫_{K_{y}} λ(k)f(e) dk
= ∫_{K_{y}} f(k^{−1}e) dk
= ∫_{K_{y}} f(ek^{−1}) dk
= ∫_{K_{y}} f(e) dk, since f ∈ M(y)
= f(e)

and if g_{0} is an inversion of y, then g_{0}y = y, so we have

(P_{λ}(y)f)(g_{0}) = ∫_{K_{y}} λ(k)f(g_{0}) dk
= ∫_{K_{y}} f(k^{−1}g_{0}) dk
= ∫_{K_{g_{0}y}} f(g_{0}h^{−1}g_{0}^{−1}g_{0}) dh
= ∫_{K_{y}} f(g_{0}h^{−1}) dh
= ∫_{K_{y}} f(g_{0}) dh, since f ∈ M(y)
= f(g_{0})

So the image of f under the action of P_{λ}(y) is the function u ∈ P_{λ}(y)(M(y)) such that ũ = f̃ on (a, b) and (b, a).

Lemma 2.4.6 Every nontrivial, closed, left-invariant subspace of M(y) contains a nontrivial, left-K_{y}-invariant function from M(y).

Proof:

Let M be our closed, non-trivial, left-invariant subspace of M(y).

Thus there is some u ∈ M, u ≠ 0. We can assume without loss of generality that u(e) ≠ 0 (since there must be some g ∈ Aut(Γ)) for which u(g) ≠ 0, and so the left translate by g^{−1} must have non-zero value at e).

Now P_{λ}(y)(M) ⊆ M, since M is left-invariant, so P_{λ}(y)u ∈ M, but from above (P_{λ}(y)u)(e) = u(e) ≠ 0, so P_{λ}(y)u is a non-trivial left-K_{y}-invariant function in M.

This means that λ_{y} can have at most 2 irreducible subrepresentations, since if there were n subrepresentations, then there would be n closed, non-trivial, left-invariant subspaces, which means that the dimension of the left-K_{y}- invariant subspace would have to be ≥ n. So there are at most 2.

Therefore either λ_{y} is irreducible, or λ_{y} = σ_{1} ⊕ σ_{2}, where σ_{1}, σ_{2} are irreducible.

Lemma 2.4.7 y is a minimal tree for λ_{y} and for any subrepresentations it might have.

Proof:

For any v ∈ V, f ∈ M(y), g ∈ Aut(Γ),

(P_{λ}(v)f)(g) = ∫_{K_{v}} f(k^{−1}g) dk
= ∫_{K_{gv}} f(gk^{−1}g^{−1}g) dk
= ∫_{K_{gv}} f(gk^{−1}) dk
= 0

So P_{λ}(v) = 0 for all v ∈ V, and from above we have shown that there are functions for which P_{λ}(y) is non-zero. Therefore y is minimal.

Thus the irreducible subrepresentations of λ_{y} are special.

Furthermore, they are essentially the only special representations, by the following lemma.

Lemma 2.4.8 Every special representation is unitarily equivalent to a subrepresentation of λ_{y}.

Proof:

If π is special, and ξ ∈ P_{π}(y)ξ, then ⟨π(g)ξ, ξ⟩ ∈ S(y) ⊆ L^{2}(Aut(Γ)). So π is square integrable, and so by the theory of square integrable representations π can be intertwined with λ, so π ≃ some subrepresentation of λ_{y} (see Section 2.2).

Which brings us to our final theorem for this section.

Theorem 2.4.9 λ_{y} is the direct sum of two inequivalent, irreducible subrepresentations σ_{1} and σ_{2}.

Proof:

Let f_{+}(y) = f_{−}(y) = 1 and h_{+} = 1, h_{−} = −1 and f, h ∈ the left- K_{y}-invariant subspace of M(y). This is a basis for this subspace, since (1, 1) and (1, −1) are orthogonal in C^{2}, so ⟨f, h⟩ = 0 in L^{2}(Aut(Γ)).

Let g_{0} be an inversion on y. Then λ(g_{0})f = f and λ(g_{o})h = −h.

Now consider the closed subspaces M_{1} and M_{2} generated by the sets {λ(s)f : s ∈ Aut(Γ)} and {λ(s)h : s ∈ Aut(Γ)} respectively. These two spaces are not equal to one another, and are both clearly non-trivial closed subspaces of M(y).

Thus M_{1} and M_{2} carry subrepresentations of σ_{1} and σ_{2} of λ_{y}.

Let F (t) = ⟨λ(t)f, h⟩. F is left-K_{y}-invariant, since for all k ∈ K_{y},

F (kt) = ⟨λ(kt)f, h⟩
= ⟨λ(t)f, λ(k^{−1})h⟩
= ⟨λ(t)f, h⟩

since h is K_{y}-invariant. However F_{+}(y) = F_{−}(y) = ⟨f, h⟩ = 0, which implies that F = 0.

This gives us that ⟨v, w⟩ = 0 for all v ∈ M_{1}, w ∈ M_{2}. Thus M_{1} ⊥ M_{2}, and hence M_{1} ⊕ M_{2} = M(y) (if there were any other direct summands, then they would have to be left-K_{y}-invariant, which is a contradiction).

Therefore λ_{y} = σ_{1} ⊕ σ_{2}, and they must both be irreducible.

Now assume that σ_{1} is equivalent to σ_{2}. Then there is some unitary operator U : M_{1} → M_{2} such that Uσ_{1}(g) = σ_{2}(g)U. Since σ_{1}, σ_{2} are subrepresentations of λ_{y}, so we can write Uλ_{y}(g) = λ_{y}(g)U, which implies that UP_{λ}(y) = P_{λ}(y)U, that is U maps the left-K_{y}-invariant functions of M_{1} to the left-K_{y}-invariant functions of M_{2}, which implies that Uf = ch, where c = ±1. However λ(g_{0})Uf = −ch, but Uλ(g_{0})f = Uf = ch, which implies U = 0. Thus we have a contradiction. So the two representations are not equivalent.

It can also be shown that σ_{1} and σ_{2} have formal dimensions d_{σ_{1}} = d_{σ_{1}} = \frac{q−1}{2} (see Figà-Talamanca [4, III, §2, p118]). This is useful in determining the Plancherel Formula for L^{2}(Aut(Γ)) (see Section 2.6).

2.5 Cuspidal Representations

We now move on to briefly deal with the cuspidal representations of Aut(Γ). Let π be cuspidal, and let T be its minimal tree. We have the following initial results.

Lemma 2.5.1 Let T be a complete subtree of Γ with diam(T) ≥ 2. Let S be a finite complete subtree with T ⊈ S. Then there exists a proper complete subtree Z of T such that K_{Z} ⊆ K_{S}K_{T}.

Proof:

Without loss of generality, we may assume that T ∩S contains an edge. This follows since if T ∩ S is either empty or a single edge, then there exists some unique v ∈ T of minimal distance m from S. We then set S′ = B_{S}(m + 1) = S_{m+1}. The intersection of T and S′ is clearly an edge, and if T ⊂ S′ then either T has diameter < 2 or T is not complete. Clearly K_{S′} ⊂ K_{S}.

Let Z = S ∩ T. Then Z is a complete, proper subtree of T. It is also a complete subtree of S. Hence given any k ∈ K_{Z} we can find an automorphism k_{2} ∈ K_{T} which agrees with k on S \ T. Similarly, we can find an automorphism k_{1} ∈ K_{S} such that k_{1} = kk_{2}^{−1}, since kk_{2}^{−1} clearly stabilises S. Hence k = k_{1}k_{2} and therefore K_{Z} ⊆ K_{S}K_{T}.

Proposition 2.5.2 Let T be a finite complete subtree of Γ with diam(T) ≥ 2. Let S be a finite complete subtree of Γ. If u is a left-K_{S}-invariant element of S(T), then supp(u) ⊂ {g ∈ Aut(Γ) : gT ⊆ S}.

Proof:

We want to show that if u ∈ S(T) is left-K_{S}-invariant, then u(g) = 0 for all g such that gT ⊈ S.

If g = e and T ⊈ S, then there exists a complete Z ⊂ T such that K_{Z} ⊆ K_{S}K_{T}. Since u is left-K_{S}-invariant and right-K_{T} -invariant, then u must be constant on K_{Z}. However, since Z ⊂ T, we have that for all h ∈ K_{Z},

∫_{K_{Z}} u(gk) dk = ∫_{K_{Z}} u(g) dk = 0

which means that u(h) = 0 for all h ∈ K_{Z}. In particular, this means u(e) = 0.

Now, for more general g, we consider u(kg) ∈ S(gT). u(kg) is still left-K_{S}-invariant, so by the above argument, if gT ⊈ S, then u(eg) = u(g) = 0.

Corollary 2.5.3 Let [T] = {gT : g ∈ Aut(Γ)}, then T′ is minimal for π iff T′ ∈ [T].

Proof:

We already know that if T is minimal, then gT is also minimal.

If T′ is minimal, then T′ must have ℓ_{π} vertices. Furthermore, let ξ be a non-trivial K_{T} - invariant vector, and let η be a non-trivial K_{T′}-invariant vector. Then ⟨π(.)ξ, η⟩ ∈ S(T) where T_{⟨π(.)ξ,η⟩} = T′. This function is non-trivial, and so has non-trivial support. Hence by the Proposition above, there is some g such that gT ⊆ T′. Therefore gT = T′, since the number of vertices in T, gT and T′ are the same.

Corollary 2.5.4 S(T) ⊆ L^{2}(Aut(Γ))

Proof:

If u ∈ S(T), then |u| < ∞, since if there were some g ∈ Aut(Γ) such that u(g) = ∞, then u(gk) = ∞ for all k ∈ K_{T}, which implies since K_{T} ⊂ K_{T^{∘}} for a complete proper subtree T^{∘} of T, and μ(K_{T}) > 0, that

∫_{K_{T^{∘}}} u(gk) dk = ∞

Hence, since T has compact support, and u is bounded, it must be L^{2}.

We are again interested in the subspace of left-K_{T} -invariant functions. As with the previous section, we can show that the function is determined by its value on the subgroup of automorphisms which stabilise T as a set, ie. the subgroup K̃_{T} = {g ∈ Aut(Γ) : gT = T }. It can be shown that K_{T} is normal in K̃_{T}, and since these functions are constant on K_{T} it makes sense to consider them as functions on K̃_{T} /K_{T}, which is a finite group isomorphic to Aut(T).

An argument similar to the one in the previous section will give us that the subspace of left-K_{T} -invariant functions has dimension equal to the number of elements in K̃_{T} /K_{T}.

We can therefore consider S(T) as the space of functions on Γ × Aut(Γ). This space is the analogue of S̃ in the special case, and so we will call it S̃(T)

Let M(T) be the subspace of L^{2}(Γ) consisting of functions for which conditions (i) and (iii) of S(T) hold.

Clearly we have that S(T) ⊂ M(T), and M(T) is a non-trivial, closed, left-invariant subspace of L^{2}. Thus we can restrict λ to M(T), and we will call this restriction λ_{T}. Again, λ_{T} is unitarily equivalent to λ_{gT}. Finally P_{λ}(T′)M(T) ⊂ S(T) for all finite complete subtrees T′.

Proposition 2.5.5 M(T) is the closure of S(T) in L^{2}(Aut(Γ)).

Proof:

The proof of this involves showing that any f ∈ M(T) is the limit of some sequence f_{n} in S(T).

The proof of this is essentially the same as for the special case, but we use S̃(T) instead of S̃ (see Proposition 2.4.5).

We can again deduce the action of P_{λ}(T), since if g ∈ K̃_{T}, then

(P_{λ}(T)f)(g) = ∫_{K_{T}} λ(k)f(g) dk
= ∫_{K_{T}} f(k^{−1}g) dk
= ∫_{K_{gT}} f(gh^{−1}g^{−1}g) dh

= ∫_{K_{T}} f(gh^{−1}) dh
= ∫_{K_{T}} f(g) dh, since f ∈ M(T)
= f(g)

Thus the action of P_{λ}(T) is to take f to the left-K_{T} -invariant function which agrees with it on K̃_{T}.

We now have, by analogy with the special case, the following results

Lemma 2.5.6 Every nontrivial, closed, left-invariant subspace of M(T) contains a nontrivial, left-K_{T} -invariant function from M(T).

Lemma 2.5.7 T is a minimal tree for λ_{T} and for any subrepresentations it might have.

Lemma 2.5.8 Every special representation is unitarily equivalent to a subrepresentation of λ_{T}.

Proof: These are proved in exactly the same fashion as for the special case. (See Lemmas 2.4.6,2.4.7 and 2.4.8).

Before we can identify the irreducible subrepresentations, we need to introduce the following.

Definition 2.5.1 Let T_{i}, i = 1, …, j be the maximal complete proper subtrees of T.

A unitary representation of K̃_{T} is called non-degenerate if it has no nonzero K_{T_{i}}-invariant vectors for i = 1, …, j.

Let (K̃_{T})_{0}^{∧} be the set of all such representations which are trivial on K_{T}.

Then we have the following.

Theorem 2.5.9 If

σ_{T} = ⨁_{σ∈(K̃_{T})_{0}^{∧}}(dimσ)σ

then λ_{T} is unitarily equivalent to

Ind(σ_{T}) = ⨁_{σ∈(K̃_{T})_{0}^{∧}}(dimσ)Ind(σ)

and each of the Ind(σ) is irreducible (and therefore cuspidal).

Proof: See Theorem 3.14 of Figà-Talamanca [4, III, §3, p134].

Furthermore, it is also possible to show that for each finite complete subtree T of diameter ≥ 2, there exists a cuspidal representation π for which T is a minimal tree. (See Figà-Talamanca [4, III, §3, p124]). This is important in finding the Plancherel Formula for L^{2}(Aut(Γ)).

2.6 Extensions and Generalisations

In this section we will tie up some loose ends.

Spherical Representations

Although spherical representations cannot be dealt with in this thesis, as the required theory would add another 30 pages, the basic method of attack and the main results should be given. For complete details, the interested reader is directed to Chapter II of Figà-Talamanca [4], and the papers by Cartier [1] and Figà-Talamanca and Steger [5].

The approach is via investigation of spherical harmonic functions on Γ, that is functions f which depend only on the distance from some fixed vertex o ∈ V, and in addition have the property that

Lf(u) = (q + 1)^{−1} ∑_{v∈∂B_{o}(1)} f(u) = cf(u)

for some constant c (L is called the Laplace operator on Γ, and hence f is an eigenfunction of L), and finally f(o) = 1.

It can be shown that the spherical representations of Aut(Γ) correspond to the positive definite spherical functions, and that they have the form π_{z} : G → L^{2}(Ω, ν_{o}) where ⟨π_{z}1, 1⟩ is the positive definite spherical function with associated eigenvalue (q + 1)^{−1}q^{1/2}2Re(q^{it}) where z = \frac{1}{2} + it for t ∈ [0, π/ ln(q)], and 1 is the function which is 1 everywhere on Ω.

From this it is possible to get that for all left-K_{o}-invariant functions f on AutΓ with compact support, we have

∥f∥^{2} = ∫_{J} ∥π_{\frac{1}{2} +it}(f)∥_{HS}^{2} dm(t)

where J = [0, π/ ln(q)] and dm(t) = \frac{q ln(q)}{2π(q+1)}|c(\frac{1}{2} + it)|^{−2}dt, where c(z) = \frac{1}{q+1} \frac{q^{1−z}−q^{z−1}}{q^{−z}−q^{z−1}} , and ∥.∥_{HS} is the Hilbert-Schmidt norm defined on the bounded linear operators on the appropriate Hilbert space, where ∥T ∥_{HS} = Tr(T^{∗}T)^{1/2}.

The Plancherel Formula for L^{2}(Aut(Γ))

Again we will just state the basic result, which is that for every continuous function f on Aut(Γ) with compact support, we have:

∥f∥_{2}^{2} = ∫_{J} ∥π_{\frac{1}{2} +it}(f)∥_{HS}^{2} dm(t) + \frac{q − 1}{2} ∑_{i=1} 2(Tr(σ_{i}(f)σ_{i}(f)^{∗})

+ ∑_{[T],diam(T)≥2} \frac{1}{μ(K̃_{T})}( ∑_{σ∈(K̃_{T})_{0}^{∧}} dim σTr(Ind(σ)(f)(Ind(σ)(f))^{∗}))

For the reasoning behind this, see Figà-Talamanca [4, III, §3, p136].

Other Groups

Much of this chapter holds for any closed subgroup G of Aut(Γ) which can be written in the form G = KF = F K where K is compact and F is a transitive subgroup for which g^{−1}Kg∩F = {e}, since then G is unimodular, and it will have most of the required properties. However, it also needs to have a sufficiently large rotation group for the section on cuspidal representations to hold.

The two most interesting such subgroups would be Aut_{+}(Γ) which is the subgroup of Aut(Γ) generated by all rotations (ie. it is inversion-free), and P SL(2, P), where P is a p-adic field (which acts on a homogeneous tree, see Figà-Talamanca [4, Appendix, §5, p156]. Both these groups are inversion free which means that there is only one irreducible special representation. In the case of Aut_{+}(Γ) the method for the cuspidal representations still works, however, P SL(2, P) has insufficient rotations.

Again, for more information, see Chapter III of Figà-Talamanca [4].

Bibliography

[1] P. Cartier, Harmonic Analysis on Trees, Proc. Symp. Pure Math. A.M.S., 26 (1972), pp419–424.

[2] J. Dixmier^{1}, Les C^{∗}-algebras et Leurs Representations, Gauthier-Villars, Paris, 1964.

[3] R.E. Edwards, Integration and Harmonic Analysis on Compact Groups, Australian National University, Canberra, 1970.

[4] A. Figà-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, Manuscript.

[5] A. Figà-Talamanca and T. Steger, Harmonic Analysis on Trees, Symp. Math., 29 (1987), pp163–182.

[6] P.R. Halmos, Measure Theory, Van Nostrand, Princeton, 1950.

[7] L.H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand, Princeton, 1953.

[8] G.I. Ol’shianskii, Classificiation of Irreducible Representations of Groups of Automorphisms of Bruhat-Tits Trees, Functional Anal. Appl., (1977), pp26–34.

[9] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, Oxford, 1968.

[10] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982.

[11] J.P. Serre, Trees, Springer-Verlag, New York and Berlin, 1980.

1Appropriate sections translated by Professor Sutherland