2.1 Graphs

An (undirected) graph $\Gamma $ is defined as a pair $(\mathrm V\Gamma , \mathrm E\Gamma )$ , where $\mathrm V\Gamma $ is the set of vertices and $\mathrm E\Gamma $ is a set of two-element subsets of $\mathrm V\Gamma $ , whose elements we call the edges of $\Gamma $ . Our graphs thus have neither loops nor multiple edges. The set of arcs of $\Gamma $ , denoted by $\mathrm A\Gamma $ , is the set of all ordered pairs $(\alpha , \beta )$ such that $\{\alpha , \beta \}\in \mathrm E\Gamma $ . Two vertices $\alpha $ and $\beta $ are called adjacent, or neighbours, if $\{\alpha , \beta \}$ is an edge. The set of neighbours of a vertex $\alpha $ is denoted by $\Gamma (\alpha )$ and the degree of $\alpha $ is the cardinality of $\Gamma (\alpha )$ . A graph is regular if all vertices have the same degree d, and then we say that d is the degree of the graph. A graph is locally finite if every vertex has finite degree.

We also consider digraphs (directed graphs). A digraph consists of a vertex set $\mathrm V\Gamma $ and a subset $\mathrm A\Gamma \subseteq \mathrm V\Gamma \times \mathrm V\Gamma $ that does not intersect the diagonal. Elements of $\mathrm V\Gamma $ are called vertices and elements of $\mathrm A\Gamma $ are called arcs. The underlying undirected graph of a digraph $\Gamma $ has the same vertex set as $\Gamma $ and the set of edges is the set of all pairs $\{\alpha , \beta \}$ where $(\alpha , \beta )$ or $(\,\beta , \alpha )$ is an arc in $\Gamma $ . If $\alpha \in \mathrm V\Gamma $ in a digraph $\Gamma $ we define the set of in-neighbours as $\in(\alpha)=\{\beta\in{\mathrm V}\Gamma\mid (\beta,\alpha)\in{\mathrm A}\Gamma\}$ and the set of out-neighbours as $\mathrm{out}(\alpha)=\{\beta\in{\mathrm V}\Gamma\mid (\alpha,\beta)\in{\mathrm A}\Gamma\}$ . The cardinality of $\mathrm{in}(\alpha )$ is the in-degree of $\alpha $ and the cardinality of $\mathrm{out}(\alpha )$ is the out-degree of $\alpha $ . A digraph is regular if any two vertices have the same in-degree and also the same out-degree.

For an integer $s \geq 0$ an s-arc in $\Gamma $ (here $\Gamma $ can be an undirected graph or a digraph) is an $(s+1)$ -tuple $(\alpha _{0},\ldots ,\alpha _{s})$ of vertices such that for every $0 \leq i \leq s-1$ the ordered pair $(\alpha _{i},\alpha _{i+1})$ is an arc in $\Gamma $ , and $\alpha _{i-1} \neq \alpha _{i+1}$ for all $1 \leq i \leq s-1$ . Infinite arcs come in three different shapes. There are one-way infinite arcs $(\alpha _{0}, \alpha _{1}, \ldots )$ and $(\ldots , \alpha _{-1}, \alpha _{0})$ , and there are two-way infinite arcs $(\ldots , \alpha _{-1}, \alpha _{0}, \alpha _{1}, \ldots )$ . In all cases we insist that $(\alpha _{i},\alpha _{i+1})$ is an arc in $\Gamma $ , and $\alpha _{i-1} \neq \alpha _{i+1}$ for all i.

A graph $\Gamma $ is said to be connected if for every pair of vertices $\alpha $ and $\beta $ in $\Gamma $ there exists an s-arc $(\alpha _{0},\ldots ,\alpha _{s})$ with $\alpha = \alpha _{0}$ and $\beta = \alpha _{s}$ . The smallest possible s is the distance between $\alpha $ and $\beta $ and is denoted by $d_{\Gamma }(\alpha , \beta )$ . A digraph is said to be connected if the underlying undirected graph is connected.

2.2 Group actions

Let $\mathrm {Sym}(\Omega )$ denote the group of all permutations of the set $\Omega $ . By a permutation group on $\Omega $ we mean a subgroup of $\mathrm {Sym}(\Omega )$ . An action of a group G on a set $\Omega $ is defined as a homomorphism $\pi : G\to \mathrm {Sym}(\Omega )$ . We write our group action on the right so that if $\alpha \in \Omega $ and $g\in G$ then $\alpha g$ denotes the image of $\alpha $ under the permutation $\pi (g)$ . The kernel of the action (that is, the kernel of the homomorphism $\pi $ ) is equal to the set $K=\{g\in G\mid \alpha g=\alpha , \mathrm{\ for\ all\ }\alpha \in \Omega \}$ . If $K=\{1\}$ , we say that the action is faithful. Set $G^{\Omega }=G/K$ . The homomorphism $\pi $ induces an injective homomorphism $G^{\Omega }\to \mathrm {Sym}(\Omega )$ giving an action of $G^{\Omega }$ on $\Omega $ . This action is clearly faithful and thus we can think of $G^{\Omega }$ as a permutation group on $\Omega $ . We call $G^{\Omega }$ the permutation group on $\Omega $ induced by the action of G. If the action $\pi $ is faithful, then $G^{\Omega }=G$ and we can think of G itself as a permutation group.

The stabilizer in G of $\alpha \in \Omega $ is the subgroup $G_{\alpha }=\{g\in G\mid \alpha g=\alpha \}$ . For a set $A\subseteq \Omega $ the pointwise stabilizer of A is defined as the subgroup

$$ \begin{align*}G_{(A)}=\{g\in G\mid \alpha g=\alpha \mathrm{\ for\ all\ }\alpha \in A\}\end{align*} $$

and the setwise stabilizer is defined as the subgroup ${G_{\{A\}}=\{g\in G\mid A g=A\}}$ .

The orbit of a point $\alpha \in \Omega $ under G is the set $\alpha G=\{\alpha g\mid g \in G\}$ , and for $S\subseteq G$ we define the S-orbit of $\alpha $ as the set $\alpha S=\{\alpha g\mid g \in S\}$ . The orbits of $G_{\alpha }$ are called suborbits of G. An action is said to be transitive if for any two points $\alpha , \beta \in \Omega $ there is an element $g\in G$ such that $\alpha g=\beta $ , that is, every orbit under G is all of $\Omega $ . A permutation group G on a set $\Omega $ is called free (or semiregular) if $G_{\alpha }=\{1\}$ for all points $\alpha \in \Omega $ and regular if it is free and transitive.

Let $\Gamma $ and $\Delta $ be graphs (or digraphs). A graph morphism (or a digraph morphism) $\varphi : \Gamma \to \Delta $ is a map $\varphi \colon \mathrm V\Gamma \to \mathrm V\Delta $ such that if $(\alpha , \beta ) \in \mathrm A\Gamma $ then $(\varphi (\alpha ),\varphi (\,\beta )) \in \mathrm A\Delta $ . If a graph morphism $\varphi :\Gamma \to \Gamma $ is bijective and $\varphi $ induces a bijective map $\mathrm A\Gamma \to \mathrm A\Gamma $ , then $\varphi $ is an automorphism of $\Gamma $ . The set of all automorphisms of $\Gamma $ is a group, the automorphism group of $\Gamma $ , denoted by $\mathrm {Aut}(\Gamma )$ . We think of $\mathrm{Aut}(\Gamma )$ as a permutation group on $\mathrm V\Gamma $ , and the action on the vertex set induces actions on the set of edges and the set of arcs. When we say that ‘a group G acts on a graph $\Gamma $ ’ we always mean an action by automorphisms. Such an action is described by a homomorphism $G\to \mathrm{Aut}(\Gamma )$ . The permutation group on the vertex set induced by G is denoted by $G^{\Gamma }$ .

A graph or a digraph $\Gamma $ is vertex-transitive if the automorphism group acts transitively on the vertex set. We say that $\Gamma $ is edge-transitive (arc-transitive, s-arc-transitive) if the automorphism group acts transitively on the edge set (the arc set, the set of s-arcs). When the automorphism group is s-arc-transitive for all s, then $\Gamma $ is said to be highly arc-transitive.

Consider now a group G that acts vertex-transitively on a graph $\Gamma $ . Let $\alpha \in \mathrm V\Gamma $ . The stabilizer $G_{\alpha }$ clearly leaves the set $\Gamma (\alpha )$ invariant and thus induces an action on it. The kernel of this action is $G_{(\Gamma (\alpha ))} \cap G_{\alpha }$ and the quotient $G_{\alpha }^{\Gamma (\alpha )}=G_{\alpha } / (G_{(\Gamma (\alpha ))} \cap G_{\alpha })$ embeds as a subgroup into $\mathrm {Sym}(\Gamma (\alpha ))$ . Now let $\alpha ^{\prime }$ be another vertex of $\Gamma $ . By assumption there exists $g \in G$ with $\alpha g = \alpha ^{\prime }$ , and $G_{\alpha ^{\prime }}=g^{-1} G_{\alpha } g$ acts on $\Gamma (\alpha ^{\prime })$ . The actions of $G_{\alpha }$ on $\Gamma (\alpha )$ and $G_{\alpha ^{\prime }}$ on $\Gamma (\alpha ^{\prime })=\Gamma (\alpha )g$ are conjugate via g and hence isomorphic. Thus, the following definition is independent of the choice of $\alpha $ .

Given a group G acting on a set $\Omega $ , we say that an equivalence relation on $\Omega $ is a G-congruence if $\alpha g$ is equivalent to $\beta g$ if and only if $\alpha $ is equivalent to $\beta $ . The orbits of a normal subgroup $N\trianglelefteq G$ form the equivalence classes of a G-congruence.

When $\sim $ denotes an equivalence relation on the vertex set of a graph (or a digraph) $\Gamma $ we can form the quotient graph (quotient digraph) $\Gamma /\!\sim $ . The vertex set of $\Gamma/\!\thicksim$ is the set of $\sim $ -classes. The arc set is defined as follows: if A and B are two $\sim $ -classes, then $(A,B)$ is an arc in $\Gamma /\!\sim $ if and only if there is a vertex $\alpha \in A$ and a vertex $\beta \in B$ such that $(\alpha ,\beta )$ is an arc in $\Gamma $ . If H is a subgroup of $\mathrm{Aut}(\Gamma )$ then $\Gamma /H$ denotes the quotient graph of $\Gamma $ with respect to the equivalence relation whose classes are the H-orbits on the vertex set. If G acts on $\Gamma $ and $\sim $ is a G-congruence then G has a natural action on the set of $\sim $ -classes that gives an action on the quotient digraph $\Gamma /\!\sim $ by automorphisms.

2.3 The permutation topology

When G is a group acting on a set $\Omega $ , for example the automorphism group of a graph $\Gamma $ acting on the vertex set $\mathrm V\Gamma $ , we can endow G with the permutation topology; see, for instance, [Reference Möller23, Reference Woess40]. One description of this topology is as follows. A neighbourhood basis of the identity element consists of all subgroups of the form $G_{(\Phi )}$ , where $\Phi $ ranges over all finite subsets of $\Omega $ . Thus a subgroup is open if and only if it contains the pointwise stabilizer of some finite subset of $\Omega $ . Another way to describe this topology is to think of $\Omega $ as having the discrete topology and then the permutation topology is the compact-open topology on G. The compact-open topology has the property that the action map $\Omega \times G\to \Omega; (\alpha , g)\mapsto \alpha g$ is continuous. If the group G already has a topology and the stabilizer $G_{\alpha }$ of a point $\alpha \in \Omega $ is open, then the permutation topology is a subset of the topology on G. If the action of G is faithful then G is totally disconnected.

Various topological properties of the permutation topology have natural descriptions in terms of the group's action. For instance, a sequence $\{g_{i}\}$ of elements in G converges to an element $g\in G$ in the permutation topology if and only if for each element $\alpha \in \Omega $ there is a number $N_{\alpha }$ such that if $i\geq N_{\alpha }$ then $\alpha g_{i}=\alpha g$ . We say that G is closed in the permutation topology if $G^{\Omega }$ is closed in $\mathrm {Sym}(\Omega )$ with respect to the permutation topology on $\mathrm {Sym}(\Omega )$ . Compactness and cocompactness have natural descriptions in the permutation topology.

Suppose that G already has a given topology and acts transitively on a set $\Omega $ and the stabilizers of points are compact open subgroups. Then any compact subset of G will also be compact in the permutation topology and thus have finite orbits on $\Omega $ .

From the above it also follows that if G is a closed subgroup of the automorphism group of a locally finite graph $\Gamma $ , then G with the permutation topology is a totally disconnected, locally compact group. The reader may also find it reassuring to know that if G is a permutation group on a countable set $\Omega $ , then the permutation topology is metrizable. Enumerate the points in $\Omega $ as $\alpha _{1}, \alpha _{2}, \ldots .$ Take two elements $g, h\in G$ . Let n be the smallest number such that $\alpha _{n} g\neq \alpha _{n} h$ or $\alpha _{n} g^{-1}\neq \alpha _{n} h^{-1}$ . Set $d(g,h)=1/2^{n}$ . Then d is a metric on G that induces the permutation topology.

2.4 Cayley–Abels graphs

Central in this work is the concept of a Cayley–Abels graph for a cgtdlc group. Recall that a fundamental theorem of van Dantzig [Reference van Dantzig35] says that every totally disconnected, locally compact group contains a compact open subgroup.

The following graph-theoretical lemma is well known but is stated here with a sketch of a proof to assist the reader.

A Cayley–Abels graph for a totally disconnected, locally compact group exists if and only if the group is compactly generated. Let $\Gamma $ be a Cayley–Abels graph for G. Suppose that $\alpha $ is a vertex and $g_{1}, \ldots , g_{d}$ are elements in G such that $\{\alpha g_{1}, \ldots , \alpha g_{d}\}=\Gamma (\alpha )$ . By Lemma 2.4, the subgroup $H=\langle g_{1}, \ldots , g_{d}\rangle $ is vertex-transitive and the compact set $G_{\alpha }\cup \{g_{1}, \ldots , g_{d}\}$ generates G. As a consequence we see that if $g_{1},\ldots , g_{k}$ are elements in G such that the set $\{\alpha g_{1}, \ldots , \alpha g_{k}\}$ contains a representative of every orbit of $G_{\alpha }$ on $\Gamma (\alpha )$ , then the set $G_{\alpha }\cup \{g_{1}, \ldots , g_{k}\}$ generates G. These facts are used repeatedly. Two different ways of constructing a Cayley–Abels graph for a cgtdlc group G are described below.

The first construction (see [Reference Krön and Möller19, Construction 1]) goes as follows. Start with the set $G/U$ of right cosets of some compact open subgroup U of G. This is the vertex set of our graph. Then choose group elements $g_{1}, \ldots , g_{n}$ such that $G=\langle U, g_{1}, \ldots , g_{n}\rangle $ and finally take some element $\alpha \in G/U$ and let the edge set be the union of the G-orbits $\{\alpha , \alpha g_{1}\}G, \ldots , \{\alpha , \alpha g_{n}\}G$ . The resulting graph is a Cayley–Abels graph for G.

The second construction is in Abels’ paper [Reference Abels1, Beispiel 5.2] (see also [Reference Krön and Möller19, Construction 2]). Start by taking a compact generating set S and a compact open subgroup U. Then construct the Cayley graph with respect to this generating set S, and finally contract the right cosets of U. After all this, we are left with a locally finite, connected graph on which G acts transitively. The stabilizers of this graph's vertices are U and its conjugates.

In both constructions the constructor has choices and thus there is always more than one possible Cayley–Abels graph. But they are all locally finite and thus the following concept is well defined.

2.5 The scale function and tidy subgroups

The pioneering paper of Willis [Reference Willis37] started a new wave of interest in totally disconnected, locally compact groups. The fundamental concepts of Willis’s theory are the scale function and tidy subgroups. In a later paper [Reference Willis38], Willis gave the following simple definitions of these fundamental concepts and showed that these new definitions are equivalent to his original definitions in [Reference Willis37].

Definition 2.7 (see [Reference Willis38, Theorem 3.1])

Let G be a totally disconnected, locally compact group. The scale function on G is defined as

$$ \begin{align*}s(g) =\min\{|U:U\cap g^{-1}Ug| \mid U \text{ compact open subgroup of } G\}.\end{align*} $$

A compact open subgroup U of G is said to be tidy for g if and only if this minimum is attained at U, that is, s(g) = |U: U ∩ g ⁻¹ Ug|.

The following proposition allows us to compute the scale function of elements by using an action of the group on a set.

Proposition 2.8 [Reference Möller22, Corollary 7.8]

Let G be a totally disconnected, locally compact group and let U be a compact open subgroup of G. Consider the action of G on the set of right cosets $\Omega =G/U$ . Set $\alpha =U$ and think of $\alpha $ as a point in $\Omega $ . If $g\in G$ then

$$ \begin{align*}s(g)=\lim_{n\to \infty} |(\alpha g^{n}) G_{\alpha}|^{1/n},\end{align*} $$

and, furthermore, $s(g)=1$ if and only if there is a constant C such that $|(\alpha g^{i})G_{\alpha }|\leq C$ for all $i=0, 1, 2, \ldots .$

5.1 Reading the modular function off the edges

The following result linking the modular function and sizes of suborbits was proved by Schlichting [Reference Schlichting29, Lemma 1]; see also [Reference Trofimov33, Theorem 1].

Lemma 5.1. Let G be a totally disconnected, locally compact group and $\Delta $ the modular function on G. If U is a compact open subgroup of G and $g\in G$ then

$$ \begin{align*}\Delta(g)=\frac{|U:U\cap g^{-1}Ug|}{|g^{-1}Ug:U\cap g^{-1}Ug|}.\end{align*} $$

Suppose that G acts on a set $\Omega $ with compact, open point stabilizers. If $\alpha \in \Omega $ and $g\in G$ then

(5-1) $$ \begin{align}\Delta(g)=\frac{|G_{\alpha}:G_{\alpha}\cap g^{-1}G_{\alpha} g|}{|g^{-1}G_{\alpha} g:G_{\alpha}\cap g^{-1}G_{\alpha} g|} =\frac{|(\alpha g) G_{\alpha}|}{|\alpha G_{ \alpha g}|}. \end{align} $$

Furthermore, the image of the modular function is a subgroup of the multiplicative group of positive rational numbers.

If G acts on a set $\Omega $ with compact point stabilizers and if for $g, h\in G$ there is an $\alpha \in \Omega $ with $\alpha g=\alpha h$ , then it follows from (5-1) that $\Delta (g)=\Delta (h)$ . If in addition G is unimodular and its point stabilizers are open, then $|(\alpha g) G_{\alpha }|=|\alpha G_{\alpha g}|$ for all g in G and all $\alpha \in \Omega $ .

The following idea originates from the paper by Bass and Kulkarni [Reference Bass and Kulkarni4, Section 3] and enables us to ‘read’ the values of the modular function for a cgtdlc group G off a Cayley–Abels graph $\Gamma $ . Label an arc $(\alpha ,\beta )$ in $\Gamma $ with the number

$$ \begin{align*}\Delta_{(\alpha, \beta)}=\frac{|\,\beta G_{\alpha}|}{|\alpha G_{\beta} |}.\end{align*} $$

Note that $\Delta _{(\,\beta , \alpha )}=\Delta _{(\alpha , \beta )}^{-1}$ . This arc labelling is clearly invariant under the action of G on the set of arcs. Combining these observations, we see that if $\Delta _{(\alpha , \beta )}\neq 1$ then there cannot exist $g\in G$ such that $(\alpha ,\beta ) g=(\,\beta ,\alpha )$ . If $g \in G$ satisfies $\alpha g=\beta $ then Lemma 5.1 readily implies that

(5-2) $$ \begin{align} \Delta(g)= \frac{|\,\beta G_{\alpha}|}{|\alpha G_{\beta}|}=\Delta_{(\alpha, \beta)}. \end{align} $$

Now let $\gamma $ be a neighbour of $\beta $ and take $h \in G$ with $\alpha h=\gamma $ . Set $g^{\prime }= g^{-1}h \in G$ . Then $\beta g^{\prime }=(\alpha g)g^{-1}h=\gamma $ . Hence, $\alpha h=\alpha g g^{\prime }$ and we see that

$$ \begin{align*}\Delta(h)=\Delta(gg^{\prime})=\Delta(g)\Delta(g^{\prime})=\Delta_{(\alpha,\beta)}\Delta_{(\,\beta,\gamma)}.\end{align*} $$

Inductively, we get that if $\alpha ,\beta \in \Gamma $ are arbitrary vertices, $g \in G$ satisfies $\alpha g=\beta $ and $(\alpha _{0},\ldots ,\alpha _{s})$ is an s-arc with $\alpha _{0}=\alpha $ and $\alpha _{s}=\beta $ , then

$$ \begin{align*}\Delta(g)=\Delta_{(\alpha_{0},\alpha_{1})}\cdot\cdots\cdot \Delta_{(\alpha_{s-1},\alpha_{s})}.\end{align*} $$

Thus the labelled Cayley–Abels graph completely describes the modular function on G. We have now proved the following theorem.

Corollary 5.3. Let G be a cgtdlc group, $\Delta $ the modular function on G and $\Gamma $ a Cayley–Abels graph for G. For a vertex $\alpha $ in $\Gamma $ , let $B_{1},\ldots ,B_{n}$ be the orbits of $G_{\alpha }$ on the neighbours of $\alpha $ and choose, for each i, an element $g_{i} \in G$ with $\alpha g_{i} \in B_{i}$ . Then

$$ \begin{align*} \mathrm{im}(\Delta) = \bigg\langle \frac{|(\alpha g_{1}) G_{\alpha}|}{|(\alpha g_{1}^{-1}) G_{\alpha}|}, \ldots, \frac{|(\alpha g_{n}) G_{\alpha}|}{|(\alpha g_{n}^{-1}) G_{\alpha}|} \bigg\rangle \leq \mathbb{Q}^{+}. \end{align*} $$

The image of $\Delta $ is a finitely generated, free abelian subgroup of the multiplicative subgroup of the positive rational numbers.

5.2 Minimal degree and the modular function

The observations in the previous subsection relating the modular function and Cayley–Abels graphs allow us to get a lower bound on the minimal degree of a Cayley–Abels graph in terms of the values of the modular function.

Theorem 5.4. Let G be a noncompact, cgtdlc group. Let $H \leq \mathbb {Q}^{+}$ be the image of the modular function on G. Then every Cayley–Abels graph of G has degree at least min(A), where

$$ \begin{align*}A = \bigg\{\sum_{i=1}^{k}(p_{i}+q_{i}) \ \bigg| \ p_{1},\ldots,p_{k},q_{1},\ldots,q_{k} \in \mathbb{N}^{*}, \, \bigg \langle \frac{p_{1}}{q_{1}}, \ldots, \frac{p_{k}}{q_{k}} \bigg \rangle=H\bigg\}\subseteq \mathbb{N}^{*}.\end{align*} $$

In particular, if H is cyclic and generated by p/q, where p, q ∈ ℕ ^* are relatively prime, then md(G) ≥ p + q.

Proof. Let $\Gamma $ be a Cayley–Abels graph for G and let $\alpha \in \mathrm V\Gamma $ . Think of $\Gamma $ as a labelled digraph as in Section 5.1. The group G has finitely many orbits E ₁, …, E _n on its edges. By vertex-transitivity, we can take an edge $e_{i}=\{\alpha , \beta _{i}\}\in E_{i}$ and set $p_{i}=|\,\beta _{i} G_{\alpha }|$ and $q_{i}=|\alpha G_{\,\beta _{i}}|$ . The label of the arc $(\alpha , \beta _{i})$ is $p_{i}/q_{i}$ . These numbers are independent of the choice of the representative $e_{i}$ of $E_{i}$ . Suppose that $g_{i}\in G$ is such that $\alpha g_{i}=\beta _{i}$ . Set $\gamma _{i}=\alpha g_{i}^{-1}$ . Then $q_{i}=|\alpha G_{\,\beta _{i}}|=|(\alpha g_{i}^{-1})G_{\alpha }|=|\gamma _{i} G_{\alpha }|$ and the label of the arc $(\alpha , \gamma _{i})$ is $q_{i}/p_{i}$ . If $p_{i}\neq q_{i}$ then the arcs $(\alpha , \beta _{i})$ and $(\alpha , \gamma _{i})$ have different labels and thus belong to different orbits of G. Therefore, the $G_{\alpha }$ -orbits $\beta _{i} G_{\alpha }$ and $(\alpha g_{i}^{-1}) G_{\alpha }$ are distinct and disjoint and are both contained in $\Gamma (\alpha )$ . The contribution of $E_{i}$ to the degree of $\alpha $ is at least $p_{i}+q_{i}$ . Hence, the degree of $\Gamma $ is at least equal to the sum of all $p_{i}+q_{i}$ with $p_{i}\neq q_{i}$ . Corollary 5.3 now says that

$$ \begin{align*}H=\bigg\langle \frac{p_{1}}{q_{1}},\ldots, \frac{p_{n}}{q_{n}}\bigg\rangle\end{align*} $$

and the result follows.

The lower bound provided in Theorem 5.4 is far away from being sharp in general. For a unimodular group the bound provided is $2$ but the minimal degree of a unimodular group can be arbitrarily large.

Theorem 5.4 does not say anything about the minimal degrees of unimodular groups, such as the automorphism groups of regular or biregular trees. (A tree is said to be biregular, or more precisely $(d,d^{\prime })$ -biregular, if all the vertices in one part of the natural bipartition have degree d and all the vertices in the other part have degree $d^{\prime }$ .) But Theorem 5.4 can be applied to the stabilizer of an end in a regular or biregular tree. Ends of graphs are defined in Section 4, but in the special case of trees the definition has a simpler form. Two rays $(\alpha _{0},\alpha _{1},\ldots )$ and $(\,\beta _{0},\beta _{1},\ldots )$ in a tree T are said to be equivalent if and only if they have the same infinite tail, that is, there exist $k,l \geq 0$ such that $(\alpha _{k},\alpha _{k+1},\alpha _{k+2},\ldots )=(\,\beta _{l},\beta _{l+1},\beta _{l+2},\ldots )$ . An end of a tree T is an equivalence class of rays. Given an end $\omega $ of a tree, it is easy to show that for each vertex $\alpha $ in T there is a unique ray $R_{\alpha }$ in $\omega $ that has $\alpha $ as its initial vertex. It is obvious that $\mathrm {Aut}(T)$ acts on the set of ends of T. If $\omega $ is an end of T, then the stabilizer of the end $\omega $ , the group $\mathrm{Aut}(T)_{\omega }$ , is a closed subgroup of $\mathrm{Aut}(T)$ . Hence, if T is locally finite, the stabilizer in $\mathrm{Aut}(T)$ of an end is a totally disconnected, locally compact group.

Proof. Clearly (3) is a special case of (2) with $d=d^{\prime }$ and (1) follows from (2) by looking at $T_{d, 2}$ . It is left to prove (2).

Denote by $V_{1}$ one of the classes of the natural bipartition of $T_{d,d^{\prime }}$ . Construct a new graph T with vertex set $V_{1}$ and edge set $\{\{\alpha ,\beta \}\mid d_{T_{d,d^{\prime }}}(\alpha , \beta )=2\mathrm{\ and\ } \beta \in R_{\alpha }\}$ . It is easy to see that this new graph is isomorphic to the $((d-1)(d^{\prime }-1)+1)$ -regular tree. The end $\omega $ of $T_{d,d^{\prime }}$ corresponds to an end of T and G acts on T fixing that end. Clearly G acts vertex- and edge-transitively on T. So, T is a Cayley–Abels graph for G and $\mathrm{md}(G)\leq (d-1)(d^{\prime }-1)+1$ .

Let $\{\alpha , \beta \}$ be an edge in T such that $\beta \in R_{\alpha }$ . Suppose that $g\in G$ and $\alpha g=\beta $ . Then

$$ \begin{align*}\Delta(g)=\frac{|\,\beta G_{\alpha}|}{|\alpha G_{\beta}|}=\frac{1}{(d-1)(d^{\prime}-1)}\end{align*} $$

and by edge-transitivity this rational number generates the image of $\Delta $ . Proposition 5.3 and Theorem 5.4 say that $\mathrm{md}(G)\geq (d-1)(d^{\prime }-1)+1$ . We conclude that $\mathrm{md}(G)=(d-1)(d^{\prime }-1)+1$ .

5.3 Digraphs and the modular function

One of the first applications of the modular function to the study of graphs and their automorphism groups was by Praeger in 1991 [Reference Praeger26]. (She does not use explicitly the term modular function but defines a function in terms of sizes of suborbits as in (5-1) in Lemma 5.1.)

Let $\vec {\mathbb {Z}}$ be the digraph with vertex set $\mathbb {Z}$ and arc set $\{(i, i+1)\mid i\in \mathbb {Z}\}$ . Similarly, we define $\vec {\mathbb {Z}}^{n}$ as the digraph with vertex set $\mathbb {Z}^{n}$ , in which $((x_{1}, \ldots , x_{n}), (y_{1}, \ldots , y_{n}))$ is an arc if and only if there is $1 \leq j \leq n$ such that $y_{j}=x_{j}+1$ and $x_{i}=y_{i}$ for all $i\neq j$ . A digraph is said to have Property Z if a surjective digraph morphism $\Gamma \to \vec {\mathbb {Z}}$ exists. A fibre of a map is the preimage of a point.

Note that, since $\vec {\mathbb {Z}}$ is vertex-transitive with in- and out-degree $1$ , a digraph morphism from a connected digraph to $\vec {\mathbb {Z}}$ is uniquely determined by the image of one vertex and this image can be chosen arbitrarily.

Praeger’s theorem can be generalized to digraphs with more than one orbit on arcs. The proof resembles the proof of Praeger’s theorem found in Evans’s paper [Reference Evans10, Theorem 3.2].

The following proposition can be seen as an addendum to Theorem 5.7.

Proof. We may assume that G is closed, since replacing G with its closure in $\mathrm {Aut}(\Gamma )$ does not change the orbits on vertices or s-arcs. Let q denote the in-degree of $\Gamma $ and let p denote the out-degree. Take vertices $\alpha $ and $\beta $ in $\Gamma $ such that $(\alpha , \beta )$ is an arc. Choose some element $g\in G$ such that $\beta =\alpha g$ . The second formula for the modular function in Lemma 5.1 now says that

$$ \begin{align*}\Delta(g)=\frac{|\,\beta G_{\alpha}|}{|\alpha G_{\beta}|} = \frac{p}{q}.\end{align*} $$

Then

$$ \begin{align*}\frac{|(\alpha g^{n})G_{\alpha}|}{|\alpha G_{\alpha g^{n}}|}=\Delta(g^{n})=\Delta(g)^{n}=\bigg(\frac{p}{q}\bigg)^{n}=\frac{p^{n}}{q^{n}}.\end{align*} $$

Note that $(\alpha , \alpha g, \ldots , \alpha g^{n})$ is an n-arc in $\Gamma $ . There are $p^{n}$ distinct n-arcs in $\Gamma $ that start at $\alpha $ , and each vertex in the orbit $(\alpha g^{n})G_{\alpha }$ is the terminal vertex of such an arc. Thus $|(\alpha g^{n})G_{\alpha }|\leq p^{n}$ . But since $|(\alpha g^{n})G_{\alpha }|/|\alpha G_{\alpha g^{n}}|=p^{n}/q^{n}$ and p and q are coprime we see that $|(\alpha g^{n})G_{\alpha }|=p^{n}$ and $|\alpha G_{\alpha g^{n}}|=q^{n}$ .

If $\gamma $ is the terminal vertex of some n-arc starting at $\alpha $ then the orbit of $\gamma $ under $G_{\alpha }$ has precisely $p^{n}$ elements. The number of n-arcs starting at $\alpha $ is $p^{n}$ , and we see that $G_{\alpha }$ acts transitively on the set of n-arcs starting at $\alpha $ . As G acts transitively on $\mathrm V \Gamma $ we see that G acts transitively on the n-arcs in $\Gamma $ . Hence, $\Gamma $ is highly arc-transitive.

Every arc in the subgraph induced by the set of descendants of $\alpha $ is contained in some n-arc starting at $\alpha $ . No two distinct n-arcs starting at $\alpha $ have a common terminal vertex. Thus the subgraph induced by the set of descendants of $\alpha $ is a tree.

In [Reference Möller22] the scale function and tidy subgroups are analysed by using graph-theoretical concepts. A prominent role in this analysis is played by highly arc-transitive digraphs such that the subgraph induced by the set of descendants of a vertex is a tree. It is therefore not surprising that the above proposition can be interpreted as a result about tidy subgroups.

Proof. We define a digraph $\Gamma $ such that the vertex set is the set of cosets $G/U$ and the set of arcs is the G-orbit $(\alpha , \beta )G$ where $\alpha =U$ and $\beta =Ug$ . Note that $G_{\alpha }=U$ . Then $\Gamma $ is an arc- and vertex-transitive digraph. The out-degree is equal to $p=|U:U\cap g^{-1}Ug|$ and the in-degree is equal to $q=|U:U\cap gUg^{-1}|$ . Since the in- and out-degrees are coprime, by Proposition 5.11 the digraph $\Gamma $ is highly arc-transitive and the subgraph induced by the set of descendants of a vertex is a tree. We also see that

$$ \begin{align*}|U:U\cap g^{-n}Ug^{n}|=|(\alpha g^{n})G_{\alpha}|=p^{n}=|U:U\cap g^{-1}Ug|^{n}.\end{align*} $$

Now [Reference Möller22, Corollary 3.5] says precisely that U is tidy for g.

The following proposition is an n-dimensional version of Proposition 5.11 and can be proved in a similar way.

6.1 The profinite Jordan–Hölder theorem

A composition series for a second countable, profinite group G is a countable descending subnormal series

$$ \begin{align*} G = G_{0} \triangleright G_{1} \triangleright G_{2} \triangleright \cdots \end{align*} $$

consisting of closed subgroups such that $\bigcap _{i \geq 0} G_{i} = \{1\}$ and such that each composition factor $G_{i-1}/G_{i}$ is simple. The number of times that a composition factor appears, up to isomorphism, is called its multiplicity. The multiplicity can be finite or countably infinite. It is well known that every profinite group has a neighbourhood basis consisting of open, normal subgroups. It is also well known that closed subgroups and Hausdorff quotients of profinite groups are again profinite. These two facts together imply that the $G_{i}$ are open subgroups of G and the composition factors are finite. By [Reference Wilson39, Lemma 0.3.1(h)] the $G_{i}$ form a neighbourhood basis of the identity.

For finite groups, composition series and composition factors are intimately tied with the classical Jordan–Hölder theorem. An analogue of this theorem holds for profinite groups. This is well known to experts, but due to the lack of a suitable reference we give a proof here.

A refinement of a subnormal series is a series that contains each subgroup of the original series. In particular, the series $G \triangleright \{1\}$ and $G \triangleright G \triangleright \{1\}$ are refinements of each other.

Proof. The first part can be proven verbatim as in the finite case, together with the classical fact that G has a neighbourhood basis of the identity consisting of open, normal subgroups.

For the second part, let $G=G_{0} \triangleright G_{1} \triangleright G_{2} \triangleright \cdots $ and $G=H_{0} \triangleright H_{1} \triangleright H_{2} \triangleright \cdots $ be two composition series for G. Let A be a finite simple group. Let $n_{1}, n_{2} \in \mathbb {N} \cup \{\infty \}$ be the multiplicities of A in the first and the second composition series, respectively. By symmetry, it is enough to show that $n_{2} \geq n_{1}$ . Let $n \in \mathbb {N}$ with $n \leq n_{1}$ . Choose $k \geq 0$ such that A appears at least n times as a quotient in the series $G_{0} \triangleright \cdots \triangleright G_{k}$ . Recall that since $G_{k}$ is open, it has finite index in G, so it has finitely many G-conjugates and $N= \bigcap _{g \in G} g^{-1}G_{k} g$ is an open, normal subgroup of G. Now recall that the $H_{i}$ form a neighbourhood basis of the identity, so there exists $\ell \geq 1$ such that $H_{\ell } \leq N$ . Now setting $N^{\prime } = \bigcap _{g \in G} g^{-1}H_{\ell } g$ , we get two finite subnormal series:

$$ \begin{align*} G &= G_{0} \triangleright G_{1} \triangleright G_{2} \triangleright \cdots \triangleright G_{k} \triangleright N \triangleright H_{1} \cap N \triangleright \cdots \triangleright H_{\ell} \cap N = H_{\ell} \triangleright N^{\prime}, \\ G &= H_{0} \triangleright H_{1} \triangleright H_{2} \triangleright \cdots \triangleright H_{\ell} \triangleright N^{\prime}. \end{align*} $$

We can refine those subnormal series so that they have simple subquotients. Let $n^{\prime }$ and $n^{\prime \prime }$ be the multiplicities of A in a composition series for the finite groups $G/N^{\prime }$ and $H_{\ell }/N^{\prime }$ , respectively. By the Jordan–Hölder theorem for finite groups and the third isomorphism theorem, A has to appear $n^{\prime }$ times as a quotient in the refinement of $G = H_{0} \triangleright H_{1} \triangleright H_{2} \triangleright \cdots \triangleright H_{\ell } \triangleright N^{\prime }$ and $n^{\prime \prime }$ times in the refinement of $H_{\ell } \triangleright N^{\prime }$ . But note that $n_{2} \geq n^{\prime }-n^{\prime \prime } \geq n$ . Since $n \leq n_{1}$ was arbitrary, we are done.

By the above theorem the following concept is well defined.

6.2 The local simple content

Let G be a totally disconnected, locally compact group. We are interested in the simple groups that appear as composition factors of every open subgroup of G. The following definition is inspired by Glöckner’s concept of the local prime content; see [Reference Glöckner12], and the work of Caprace et al. in [Reference Caprace, Reid and Willis8].

The following lemma allows us to detect the local simple content by only looking at one composition series of one compact, open subgroup.

Proof. First we prove that (1) implies (3). Assume that A is in the local simple content of G. Let U be a compact open subgroup of G and let $U = U_{0} \triangleright U_{1} \triangleright \cdots $ be a composition series for U. Then, for every $k \geq 0$ , the series $U_{k} \triangleright U_{k+1} \triangleright U_{k+2} \triangleright \cdots $ is a composition series for $U_{k}$ and A has to appear as composition factor. Consequently, A appears as composition factor of U infinitely often.

It is obvious that (3) implies (2).

To show that (2) implies (1), assume that U is a compact, open subgroup of G and $U = U_{0} \triangleright U_{1} \triangleright \cdots $ is a composition series for U such that A appears as composition factor infinitely often. Let V be a compact open subgroup of G and $V=V_{0} \triangleright V_{2} \triangleright \cdots $ be a composition series for V. Since the $V_{i}$ form a neighbourhood basis of the identity, there exists $m_{0} \geq 1$ with $V_{m_{0}} \leq U_{0}$ . Now the series

$$ \begin{align*} V=V_{0} \triangleright \cdots \triangleright V_{m_{0}} = V_{m_{0}} \cap U_{0} \triangleright V_{m_{0}} \cap U_{1} \triangleright V_{m_{0}} \cap U_{2} \triangleright \cdots \end{align*} $$

is a new composition series for V. By the Jordan–Hölder theorem for profinite groups, it has the same composition factors as the original series for V. But the $U_{i}$ form a neighbourhood basis of the identity, so for large n the equality $V_{m_{0}} \cap U_{n} = U_{n}$ holds. Since A appears as quotient of the subnormal series $U_{n} \triangleright U_{n+1} \triangleright \ldots ,$ we see that A is a composition factor of V.

The following considerations rely heavily on the proof of [Reference Caprace, Reid and Willis8, Proposition 4.6]. A subquotient of a group H is a quotient of a subgroup of H. Let G be a cgtdlc group and let $U \leq G$ be a compact, open subgroup. Let $\Gamma $ be a Cayley–Abels graph for G such that $U=G_{\alpha }$ for some vertex $\alpha \in \mathrm V\Gamma $ . We can use $\Gamma $ to produce a subnormal series for U. Namely, take subgraphs $\Gamma _{0} \subseteq \Gamma _{1} \subseteq \Gamma _{2} \subseteq \cdots $ of $\Gamma $ as follows.

1. (1) The subgraph $\Gamma _{0}$ consists only of the vertex $\alpha $ .

2. (2) For $i \geq 1$ , choose a vertex $\alpha _{i} \in \mathrm V\Gamma _{i-1}$ satisfying the following conditions.

   • • The set of neighbours of $\alpha _{i}$ is not contained in $\mathrm V\Gamma _{i-1}$ .

   • • $\alpha _{i}$ is a vertex in $\Gamma _{i-1}$ with the above property that is closest to $\alpha $ .

3. (3) Now define $\Gamma _{i}$ to be the subgraph of $\Gamma $ induced by $\Gamma _{i-1}$ and $\Gamma (\alpha _{i})$ .

Note that the second condition on $\alpha _{i}$ ensures that $\bigcup _{i\geq 0}\mathrm V\Gamma _{i}=\mathrm V\Gamma $ .

Write $G_{i} = G_{(\Gamma _{i})}$ . The sequence

$$ \begin{align*} U = G_{0} \triangleright G_{1} \triangleright G_{2} \triangleright \cdots \end{align*} $$

is a descending subnormal series for U. Note that there could be repetitions in the series. The condition $\bigcup _{i \geq 0} \mathrm V\Gamma _{i} = \mathrm V\Gamma $ implies that $\bigcap _{i \geq 0} G_{i} = K$ , where K is the kernel of the action of G on $\Gamma $ . Note that $G_{i-1}$ permutes those neighbours of $\alpha _{i}$ that are not already contained in $\Gamma _{i-1}$ and the kernel of this action is $G_{i}$ , so for $i \geq 2$ the group $G_{i-1}/G_{i}$ is a subgroup of the symmetric group $S_{d-1}$ . More precisely, let $\beta _{i} \in \mathrm V\Gamma _{i-1}$ be a neighbour of $\alpha _{i}$ ; then $G_{i-1}/G_{i}$ is a subquotient of the stabilizer of $\beta _{i}$ in the local action of G on $\Gamma $ . Quotienting every subgroup in the given subnormal series for U by K gives a subnormal series for $U/K$ , which by the Jordan–Hölder theorem for profinite groups can be refined into a composition series for $U/K$ . This provides, together with Lemma 6.5, a proof of the following theorem.

We apply this theorem to the automorphism group of a regular tree.

This argument fails for the $5$ -regular tree, because $A_{4}$ is not simple. In fact $\mathrm {Aut}(T_{4})$ and $\mathrm {Aut}(T_{5})$ have the same local simple content. The result also holds for $\mathrm {Aut}(T_{5})$ , but the proof turned out to be surprisingly tricky and is given in Section 7. The proof above also provides an alternative proof for part (1) in Corollary 5.5 for $d \neq 5$ .

6.3 The local prime content and the scale function

In this section we apply Theorem 6.6 to other invariants.

The following lemma gives the connection between the local prime content and the local simple content. If G is not second countable, we do not know whether 6.2(2) still holds (see [Reference Reid27]), so in the definition of composition factors, the ‘and hence every’ part needs to be left out. This subtlety is, however, not relevant for us.

This lemma allows us to draw the following conclusion from Theorem 6.6.

The connection between the scale function and the local prime content is given in the following lemma by Glöckner.

The next lemma is contained in a more general statement in [Reference Bywaters, Glöckner and Tornier7, Theorem C(c)], but for completeness a proof is included here.

Proof. Suppose that $\Omega $ is a set on which G acts transitively such that the stabilizers of points are compact open subgroups of G, for example $\Omega = G/U$ for some compact, open subgroup $U \leq G$ . Since K is compact in the given topology on G it is also compact in the permutation topology on G constructed from the action on $\Omega $ and thus K has finite orbits on $\Omega $ ; see the discussion after Lemma 2.2. Since K is normal, G has an action on $\Omega /K$ and the stabilizers in G of points in $\Omega /K$ are compact open subgroups of G. Let $\alpha $ be a point in $\Omega /K$ . Note that the homomorphism $G \to \mathrm {Sym}(\Omega /K)$ factors through $G/K$ , thus $(\alpha (gK))(G/K)_{\alpha } = (\alpha g)G_{\alpha }$ for every $g \in G$ . Now, by applying Proposition 2.8 to the action of $G/K$ on $\Omega /K$ and to the action of G on $\Omega /K$ , we get

$$ \begin{align*}s_{G/K}(gK)=\lim_{n\to\infty} |(\alpha (gK)^{n})(G/K)_{\alpha}|^{1/n} =\lim_{n\to\infty} |(\alpha g^{n})G_{\alpha}|^{1/n}=s(g). \\[-40pt] \end{align*} $$

We are now in a position to derive a lower bound for the minimal degree in terms of the scale function.

The following is a simple application of the above results.