Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and $\ell^2$-Multiplicities

TL;DR

This paper introduces a new convergence concept for finite simplicial complexes with group actions, defining $\, ext{ extltilde}\,$-homology and multiplicities that generalize $\, ext{ extltilde}\,$-Betti numbers, and studies their continuity and induction properties.

Contribution

It develops a novel equivariant Benjamini-Schramm convergence framework for simplicial complexes with group actions, extending $\, ext{ extltilde}\,$-homology and multiplicity concepts.

Findings

$\, ext{ extltilde}\,$-multiplicities are continuous on sofic random rooted complexes.

The framework generalizes classical $\, ext{ extltilde}\,$-Betti numbers.

Induction affects $\, ext{ extltilde}\,$-multiplicities in a predictable manner.

Abstract

We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a corresponding $\ell^2$-homology and the $\ell^2$-multiplicity of an irreducible representation of G in the homology. The $\ell^2$-multiplicities generalize the $\ell^2$-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study induction of random rooted complexes and discuss the effect on $\ell^2$-multiplicities.

Equations

$$m_n^{(2)}(\rho ,{\mu }_K)=\frac{m(\rho ,H_n(K,\mathbb{C}))}{|K^{(0)}|};$$

$$\underset{k\to \infty }{\lim }{m}_n^{(2)}(\rho ,{\mu }_k)=m_n^{(2)}(\rho ,{\mu }_{\infty })$$

$$m_n^{(2)}(\rho ,{\mathrm{ind}}_1^G(\mu ))=\frac{{\dim }_{\mathbb{C}}(\rho )}{|G|}{b}_n^{(2)}(\mu ).$$

$$d([K,o],[L,o^{\prime }])=\inf \left\{\frac{1}{2^r}|B_r(K,o)\stackrel{G}{\cong }{B}_r(L,o^{\prime })\right\}$$

$${\mathcal{SC}}_{\ast }^D(G)=\underset{X\in \mathrm{Orb}(G)}{⨆}{\mathcal{SC}}_{\ast }^D(G,X),$$

$${\int }_{{\mathcal{SC}}_{\ast }^D(G)}\sum _{x\in K^{(0)}}f([K,o,Gx])d\mu ([K,o])={\int }_{{\mathcal{SC}}_{\ast }^D(G)}\sum _{x\in K^{(0)}}f([K,Gx,o])d\mu ([K,o])$$

$${\int }_{{\mathcal{SC}}_{\ast }^D(G)}\sum _{\begin{array}{c}x\in K^{(0)}\\ d(x,o)\le n\end{array}}f([K,o,Gx])d\mu ([K,o])={\int }_{{\mathcal{SC}}_{\ast }^D(G)}\sum _{\begin{array}{c}x\in K^{(0)}\\ d(x,o)\le n\end{array}}f([K,Gx,o])d\mu ([K,o])$$

$${\mu }_K^G:=\sum _{x\in K^{(0)}}\frac{{\delta }_{[K,Gx]}}{|K^{(0)}|},$$

$${\int }_{{\mathcal{SC}}_{\ast }^D(G)}\sum _{x\in L^{(0)}}f([L,o,Gx])d{\mu }_K^G([L,o])$$

$$=\sum _{x\in K^{(0)}}\frac{1}{|K^{(0)}|}\sum _{y\in K^{(0)}}f([K,Gy,Gx])$$

$$={\int }_{{\mathcal{SC}}_{\ast }^D(G)}\sum _{y\in L^{(0)}}f([L,Gy,o])d{\mu }_K^G([L,o]).$$

$$G{\times }_{H}Y=(G\times Y)/\sim$$

$$d\bigl{(}\operatorname{ind}_{H}^{G}([K,o]),\operatorname{ind}_{H}^{G}([L,o^{\prime}])\bigr{)}\leq d([K,o],[L,o^{\prime}])$$

$${\int }_{{\mathcal{SC}}_{\ast }^D(G)}t(z)d{\mathrm{ind}}_H^G(\mu )(z)={\int }_{{\mathcal{SC}}_{\ast }^D(H)}t({\mathrm{ind}}_H^G(w))d\mu (w)$$

$${\int }_{{\mathcal{SC}}_{\ast }^D(G)}\sum _{x\in L^{(0)}}$$

$$\stackrel{(1)}{=}$$

$$=$$

$$\stackrel{\text{(2)}}{=}$$

$$\stackrel{(3)}{=}$$

$$\underset{n\to \infty }{\lim }\frac{|E(K_n,g,C)|}{|K_n^{(0)}|}=0$$

$$B_r(K_n,Hx)\cap g{B}_r(K_n,Hx)=\varnothing$$

$$\left|{\mu }_{K_n}^G(U(\alpha ,r))-{\mu }_{K_n}^H(V)\right|\le \frac{|E(K_n,2r+1)|}{|K_n^{(0)}|}<\epsilon .$$

$$T_{n+1}=T_n\cup (T_n+2^n{e}_1)\cup (T_n+2^n{e}_2).$$

$$N(k+1,\alpha )=3N(k,\alpha )+c_{\alpha }$$

$$\frac{1}{w(\Gamma )}\sum _{x\in \mathcal{F}}|{\mathrm{St}}_{\Gamma }(x){|}^{-1}{\delta }_{[K,x]}$$

$${\Gamma }_0\subseteq \bigcap _{h\in H}{C}_{\Gamma }(h).$$

$${\mu }_K^H=\frac{1}{w({\Gamma }_0)}\sum _{x\in {\mathcal{F}}_0}|{\mathrm{St}}_{{\Gamma }_0}(x){|}^{-1}{\delta }_{[K,Hx]}$$

$$\sum _{x\in {\mathcal{F}}_1}|{\mathrm{St}}_{{\Gamma }_1}(x){|}^{-1}{\delta }_{[K,Hx]}=\sum _{y\in {\mathcal{F}}_0}|{\mathrm{St}}_{{\Gamma }_0}(y){|}^{-1}\sum _{\begin{array}{c}\gamma \in {\Gamma }_0\\ \gamma y\in {\mathcal{F}}_1\end{array}}|{\mathrm{St}}_{{\Gamma }_1}(\gamma y){|}^{-1}{\delta }_{[K,H\gamma y]}$$

$$=$$

$${\int }_{{\mathcal{SC}}_{\ast }^D(H)}\sum _{y\in L^{(0)}}$$

$$=\frac{1}{w({\Gamma }_0)}\sum _{x\in {\mathcal{F}}_0}|{\mathrm{St}}_{{\Gamma }_0}(x){|}^{-1}\sum _{y\in K^{(0)}}f(K,Hx,Hy)$$

$$=\frac{1}{w({\Gamma }_0)}\sum _{x\in {\mathcal{F}}_0}|{\mathrm{St}}_{{\Gamma }_0}(x){|}^{-1}\sum _{y\in {\mathcal{F}}_0}\sum _{\gamma \in {\Gamma }_0}|{\mathrm{St}}_{{\Gamma }_0}(y){|}^{-1}f(K,Hx,H\gamma y)$$

$$=\frac{1}{w({\Gamma }_0)}\sum _{x,y\in {\mathcal{F}}_0}|{\mathrm{St}}_{{\Gamma }_0}(x){|}^{-1}|{\mathrm{St}}_{{\Gamma }_0}(y){|}^{-1}\sum _{\gamma \in {\Gamma }_0}f(K,H{\gamma }^{-1}x,Hy)$$

$$=\cdots ={\int }_{{\mathcal{SC}}_{\ast }^D(H)}\sum _{x\in L^{(0)}}f(L,Hx,o)d{\mu }_K^H([L,o])\text{qed}$$

$$B_{r+1}(K,H{x}_0)\cap \gamma B_{r+1}(K,H{x}_0)\ne \varnothing$$

$${\mu }_K^H(U(\alpha ,r))$$

Full text

The research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 338540207.

1. Introduction

Benjamini and Schramm [3] introduced the concept of random rooted graphs as probability measures on the space of connected rooted graphs. This allowed them to define convergence – today known as Benjamini-Schramm convergence – of sequences of finite graphs and to study the corresponding limit random rooted graphs. It was realized by Aldous and Lyons [2] that Benjamini-Schramm limits of finite graphs share a useful mass-transport property, called unimodularity, which, when added to the definition of random rooted graphs, allows to extend several results from quasi-transitive graphs to this setting. These two insights provide the basis of a large number subsequent developments.

The basic idea of Benjamini-Schramm convergence is not specific to graphs and was, in particular, generalized to simplicial complexes in the work of Elek [8], Bowen [5] and one of the authors [13]. The central result of [13] (extending similar results in [8, 5]) is that suitably defined $\ell^{2}$-Betti numbers of random rooted simplicial complexes are Benjamini-Schramm continuous (on the space of sofic random rooted complexes). For graphs a stronger result was proven in [1]. Here we introduce an equivariant Benjamini-Schramm convergence for simplicial complexes with the action of a fixed finite group; to our knowledge this is a new concept even for graphs. Then we define refinements of $\ell^{2}$-Betti numbers, namely $\ell^{2}$-multiplicities, and prove a corresponding continuity result.

Statement of main results

We fix a finite group $G$ and consider the space $\mathcal{SC}_{*}^{D}(G)$ of isomorphism classes of rooted simplicial $G$-complexes of vertex degree at most $D$. Here a rooted simplicial $G$-complex $(K,o)$ consists of a simplical complex with an action of $G$ and a distinguished $G$-orbit $o$ of vertices in $K$ which touches every connected component of $G$. A random rooted simplicial $G$-complex $\mu$ is a unimodular probability measure on $\mathcal{SC}_{*}^{D}(G)$; for details we refer to Section 2. For instance, picking a root uniformly at random in a finite simplicial $G$-complex defines a random rooted $G$-complex. Limits of laws of finite simplicial $G$-complexes are called sofic; see Definition 2.4.

Let $K$ be a finite simplical $G$-complex. The group $G$ acts on $K$ and thus acts on the homology groups $H_{n}(K,\mathbb{C})$, i.e., we have a finite dimensional representation of $G$ on $H_{n}(K,\mathbb{C})$. This representation decomposes as a direct sum of irreducible representations of $G$ and every irreducible representation $\rho\in\operatorname{Irr}(G)$ occurs a finite number, say $m(\rho,H_{n}(K,\mathbb{C}))$, of times in this decomposition. The number $m(\rho,H_{n}(K,\mathbb{C}))$ is called the multiplicity of $\rho$ in $H_{n}(K,\mathbb{C})$. Drawing from this, we define $\ell^{2}$-multiplicities in Section 4; these are nonnegative real numbers $m^{(2)}_{n}(\rho,\mu)$ for every random rooted simplicial $G$-complex $\mu$ and every irreducible representation $\rho\in\operatorname{Irr}(G)$. In fact, if $\mu_{K}$ is the law of a finite $G$-complex, then

[TABLE]

see Example 4.7. Our main result is the following approximation theorem.

Theorem 1.1**.**

Let $(\mu_{k})_{k\in\mathbb{N}}$ be a sequence of random rooted simplicial $G$-complexes. If every $\mu_{k}$ is sofic and the sequence converges weakly to $\mu_{\infty}$, then

[TABLE]

for every $n\in\mathbb{N}_{0}$ and every irreducible representation $\rho$ of $G$.

Along the way (in Section 3) we investigate induced $G$-complexes. Given a subgroup $H\leq G$ and a simplicial $H$-complex $L$, it is natural to construct a simplicial $G$-complex $K=G\times_{H}L$ by inducing the action from $H$ to $G$. This can be promoted to an operation $\operatorname{ind}_{H}^{G}$ which takes random rooted $H$-complexes to random rooted $G$-complexes. We provide a criterion to decide whether a sequence of finite $G$-complexes converges to an induced random rooted $G$-complex; see Proposition 3.3. This is relevant, since the $\ell^{2}$-multiplicities of the induced random rooted complex $\operatorname{ind}_{H}^{G}(\mu)$ can be computed from the $\ell^{2}$-multiplicities of $\mu$ (provided $\mu$ is sofic). As a special case, for $H=\{1\}$, we have the following result.

Theorem 1.2**.**

Let $\mu$ be a sofic random rooted simplicial complex and let $G$ be a finite group. For all $\rho\in\operatorname{Irr}(G)$ and $n\in\mathbb{N}_{0}$ the following identity holds:

[TABLE]

In the topological setting $L^{2}$-multiplicities of CW-complexes have been defined and studied in [10]. Our results are mainly complementary to the results therein. However, for towers of finite sheeted covering spaces of simplicial complexes our results provide a different approach to [10, Theorem 1.2]. In this setting, we think that our investigation of induced complexes is useful. Combining Theorem 1.2 with Proposition 3.6 provides a new perspective on the centralizer condition which remained a bit obscure in [10, Proposition 5.5].

2. Random rooted simplicial $G$-complexes

We will use the language of simplicial complexes as introduced in many standard texts; for instance [14]. The $n$-skeleton of a simplicial complex $K$ is denoted by $K^{(n)}$. The number of $1$-simplices containing a vertex $x\in K^{(0)}$ is called the degree of $x$. The supremum over the degrees of all vertices of $K$ will be called the vertex degree of $K$. To avoid some technical issues we will consider simplicial complexes whose vertex degree is bounded above by some large constant $D>0$. For a simplicial complex $K$, a non-empty set of vertices $V\subseteq K^{(0)}$ and $r>0$ we define $B_{r}(K,V)$ to be the full subcomplex on the set of vertices of path distance at most $r$ from a vertex in $V$.

We fix a finite group $G$. A rooted simplicial $G$-complex is a pair $(K,o)$ consisting of a simplicial complex $K$ with $G$-action and a $G$-orbit $o$ of vertices of $K$ such that every connected component of $K$ contains at least one vertex in $o$. Two rooted simplicial $G$-complexes $(K,o)$, $(L,o^{\prime})$ are isomorphic if there is a simplicial isomorphism $f\colon K\to L$ such that $f(o)=o^{\prime}$ and $g\cdot f(x)=f(g\cdot x)$ for all $x\in K^{(0)}$ and $g\in G$; in this case we write $(K,o)\stackrel{{\scriptstyle G}}{{\cong}}(L,o^{\prime})$.

We denote by $\mathcal{SC}_{*}^{D}(G)$ the space of isomorphism classes of rooted simplicial $G$-complexes with vertex degree bounded by $D$. The following defines a metric on $\mathcal{SC}_{*}^{D}(G)$:

[TABLE]

for $[K,o],[L,o^{\prime}]\in\mathcal{SC}_{*}^{D}(G)$ Note that subcomplexes of the form $B_{r}(K,o)$ are stable under the $G$-action. With the induced topology $\mathcal{SC}_{*}^{D}(G)$ is a compact totally disconnected space. For every finite rooted simplicial $G$-complex $\alpha$ of radius at most $r$, we obtain an open subset $U(\alpha,r)\subseteq\mathcal{SC}_{*}^{D}(G)$ consisting of all isomorphism classes of rooted simplicial $G$-complexes $[K,o]$ such that $B_{r}(K,o)\stackrel{{\scriptstyle G}}{{\cong}}\alpha$. The sets $U(\alpha,r)$ are open and compact and provide a basis for the topology.

Given two rooted simplicial $G$-complexes $(K,o)$ and $(L,o^{\prime})$, such that $o$ and $o^{\prime}$ are not isomorphic as $G$-sets, then $d([K,o],[L,o^{\prime}])=\infty$. Let $\operatorname{Orb}(G)$ denote the finite set of isomorphism classes of transitive $G$-sets. The space $\mathcal{SC}_{*}^{D}(G)$ splits into a disjoint union

[TABLE]

where $\mathcal{SC}_{*}^{D}(G,X)$ is the subspace of all $[K,o]$ where $o$ belongs to $X$.

In the same manner we define the space $\mathcal{SC}_{**}^{D}(G)$ of isomorphism classes of doubly rooted simplicial $G$-complexes $(K,o,o^{\prime})$ consisting of a simplicial $G$-complex $K$ and two orbits $o$ and $o^{\prime}$ of vertices which touch every connected component.

Definition 2.1**.**

A random rooted simplicial $G$-complex is a unimodular probability measure $\mu$ on $\mathcal{SC}_{*}^{D}(G)$, where unimodular means

[TABLE]

for all Borel measurable functions $f\colon\mathcal{SC}_{**}^{D}(G)\to\mathbb{R}_{\geq 0}$.

Remark 2.2**.**

A weak limit of random rooted simplicial $G$-complexes is again a random rooted simplicial $G$-complex. This can be deduced by observing that a probability measure $\mu$ on $\mathcal{SC}_{*}^{D}(G)$ is unimodular if and only if for all $n\in\mathbb{N}$ and all continuous functions $f\colon\mathcal{SC}_{**}^{D}(G)\to\mathbb{R}$ the equality

[TABLE]

holds, where the sum now runs over the vertices of path distance at most $n$ from the root orbit. This can be verified using the theorem of monotone convergence and the monotone class theorem.

Example 2.3**.**

Let $K$ be a finite simplicial $G$-complex. Then there is a unique random rooted simplicial $G$-complex which is fully supported on isomorphism classes of the form $[K,o]$; it is given by

[TABLE]

where $\delta_{[K,Gx]}$ denotes the Dirac measure of the point $[K,Gx]$. That $\mu^{G}_{K}$ is a probability measure is obvious, we only have to verify that it is unimodular:

[TABLE]

Definition 2.4**.**

A sequence $(K_{n})_{n}$ of finite simplicial $G$-complexes (of vertex degree bounded by $D$) converges Benjamini-Schramm if the weak limit $\lim_{n\to\infty}\mu_{K_{n}}^{G}$ exist. A random rooted simplicial $G$-complex on $\mathcal{SC}_{*}^{D}(G)$ which is the Benjamini-Schramm limit of a sequence of finite simplicial $G$-complexes is called sofic.

3. Induction of simplicial complexes

Let $G$ be a finite group and let $H\leq G$ be a subgroup. Given an $H$-set $Y$ one can construct the induced $G$-set $G\times_{H}Y$. The set

[TABLE]

is obtained by forming the quotient of $G\times Y$ under the equivalence relation $(g,y)\sim(gh,h^{-1}\cdot y)$ for all $g\in G$, $y\in Y$ and $h\in H$. We write $\lfloor g,y\rfloor$ to denote the equivalence class of $(g,y)$. Clearly $g_{2}\cdot\lfloor g_{1},y\rfloor=\lfloor g_{2}g_{1},y\rfloor$ defines an action of $G$ on $G\times_{H}Y$.

A simplicial $H$-complex $K$ gives rise to a simplicial $G$-complex $G\times_{H}K$ by induction. The vertices of $G\times_{H}K$ are the elements of $G\times_{H}K^{(0)}$. For every simplex $\sigma\subseteq K^{(0)}$ of $K$ and every $g\in G$ we define a simplex $\sigma_{g}=\{\lfloor g,x\rfloor\mid x\in\sigma\}$ of $G\times_{H}K$. As a simiplicial complex (without $G$-action) $G\times_{H}K$ is isomorphic to a disjoint union of $|G/H|$ copies of $K$. In particular, induction does not alter the vertex degree.

Lemma 3.1**.**

The function $\operatorname{ind}_{H}^{G}\colon\mathcal{SC}_{*}^{D}(H)\to\mathcal{SC}_{*}^{D}(G)$ which maps $[K,o]$ to $[G\times_{H}K,G\lfloor 1,o\rfloor]$ is continuous. In particular, the push-forward of measures with $\operatorname{ind}_{H}^{G}$ is weakly continuous.

Proof.

We observe that the ball of radius $r$ in $G\times_{H}K$ around $G\lfloor 1,o\rfloor$ is isomorphic to $G\times_{H}B_{r}(K,o)$. This implies that

[TABLE]

and proves the assertion. ∎

We use this to define induction of random rooted simplicial complexes.

Lemma 3.2**.**

Let $\mu$ be a random rooted simplicial $H$-complex. The push-forward measure $\operatorname{ind}_{H}^{G}(\mu)$ is a random rooted simplicial $G$-complex.

Proof.

The push-forward preserves the total mass, so $\operatorname{ind}_{H}^{G}(\mu)$ is a probability measure. It remains to verify unimodularity. Recall the general transformation rule [4, §6]

[TABLE]

for all measurable nonnegative functions $t$ on $\mathcal{SC}_{*}^{D}(G)$. We obtain for all measurable $f\colon\mathcal{SC}_{**}^{D}(G)\to\mathbb{R}_{\geq 0}$

[TABLE]

where we use the transformation rule in steps (1) and (3), and the unimodularity of $\mu$ in step (2). ∎

The following criterion is useful to show that a sequence of finite simplicial $G$-complexes converges to an induced random rooted simplicial $G$-complex.

Proposition 3.3**.**

Let $(K_{n})_{n}$ be a sequence of finite simplicial $G$-complexes with vertex degree bounded by $D$. Assume that the sequence of random rooted simplical $H$-complexes $(\mu_{K_{n}}^{H})_{n}$ converges to a random rooted simplicial $H$-complex $\mu_{\infty}$ for some subgroup $H\leq G$. Then $(\mu_{K_{n}}^{G})_{n}$ converges to $\operatorname{ind}_{H}^{G}(\mu_{\infty})$ on $\mathcal{SC}_{*}^{D}(G)$ if and only if

[TABLE]

for all $C>0$ and all $g\in G\setminus H$ where $E(K,g,C)=\{x\in K^{(0)}\mid d(x,gx)\leq C\}$.

Proof.

Define $E(K,C)=\bigcup_{g\in G\setminus H}E(K,g,C)$.

Assume that equation (3.1) holds for all $g\in G\setminus H$ and all $C>0$. Let $r>0$ be given. We want to verify that for all $x\in K_{n}^{(0)}\setminus E(K_{n},2r+1)$ the ball of radius $r$ around $Gx$ in $K_{n}$ is isomorphic to $G\times_{H}B_{r}(K_{n},Hx)$. This amounts to showing that

[TABLE]

for all $g\in G\setminus H$ and that there is no edge between these two sets. Suppose that there is an element in the intersection or an edge between $B_{r}(K_{n},Hx)$ and $gB_{r}(K_{n},Hx)$. In both cases we can find $h,h^{\prime}\in H$ such that $d(hx,gh^{\prime}x)\leq 2r+1$. However this implies $x\in E(K_{n},2r+1)$ since $d(x,h^{-1}gh^{\prime}x)\leq 2r+1$ and $h^{-1}gh^{\prime}\not\in H$.

Let $\alpha$ be a rooted simplicial $G$-complex of radius at most $r$. Let $\varepsilon>0$ and take $n$ sufficiently large such that $|E(K_{n},2r+1)|<|K_{n}^{(0)}|\varepsilon$. The inverse image $V=(\operatorname{ind}_{H}^{G})^{-1}(U(\alpha,r))\subseteq\mathcal{SC}_{*}^{D}(H)$ is a finite (possibly empty) union of sets of the form $U(\alpha^{\prime},r)$; thus it is open and compact. The weak convergence $\mu_{K_{n}}^{H}\stackrel{{\scriptstyle w}}{{\longrightarrow}}\mu_{\infty}$ shows that for all sufficiently large $n$ the inequality $|\mu_{\infty}(V)-\mu_{K_{n}}^{H}(V)|<\varepsilon$ holds. Moreover, by the observation above

[TABLE]

As $\alpha$ was arbitrary, we deduce the convergence $\mu_{K_{n}}^{G}\stackrel{{\scriptstyle w}}{{\longrightarrow}}\operatorname{ind}_{H}^{G}(\mu_{\infty})$.

Conversely, suppose that the sequence $(\mu_{K_{n}}^{G})_{n}$ converges to $\operatorname{ind}_{H}^{G}(\mu_{\infty})$. Let $g\in G\setminus H$, $C>0$ and $x\in E(K_{n},g,C)$. The ball $B_{C}(K_{n},Hx)$ contains a path from $x$ to $gx$ and hence it is not isomorphic to a simplicial complex induced from $H$. By assumption the limit $\lim_{n\to\infty}\mu_{K_{n}}^{G}$ is supported on induced complexes and thus (3.1) is satisfied. ∎

Example 3.4** (Sierpinski’s triangle with rotation).**

We describe a sequence $(T_{n})_{n}$ of two-dimensional simplicial complexes which occur in the construction of the fractal Sierpinski triangle. It appeared to us that the example becomes clearer if we describe the geometric realizations of the $T_{n}$ as subsets of $\mathbb{R}^{2}$ instead of working with the abstract simplicial complexes. Let $e_{1}=(1,0)\in\mathbb{R}^{2}$ and let $e_{2}=\frac{1}{2}(1,\sqrt{3})\in\mathbb{R}^{2}$. The points [math], $e_{1}$ and $e_{2}$ are the vertices of an equilateral triangle $T_{0}$ with sides of length $1$; we consider $T_{0}$ to be a $2$-simplex. We define inductively

[TABLE]

By induction it is easy to verify that $T_{n}$ is a simplicial complex with $\frac{3^{n+1}+3}{2}$ vertices, $3^{n+1}$ edges and $3^{n}$ $2$-simplices. The vertex degree of $T_{n}$ is $4$ for all $n\geq 1$. The three vertices of degree $2$ will be called the corners of $T_{n}$. The distance between two corners of $T_{n}$ is $2^{n}$.

Claim: The sequence $(T_{n})_{n}$ converges Benjamini-Schramm to a random rooted simplicial complex $\tau_{S}$.

Let $r>0$ and let $\alpha$ be a finite rooted simplicial complex of radius at most $r$. Take $m$ so large that $2^{m-1}>r$. Then any $r$-ball in $T_{m}$ contains at most one of the three corners of $T_{m}$. Let $N(k,\alpha)$ denote the number of vertices $v$ in $T_{m+k}$ such that the ball of radius $r$ around $v$ is isomorphic to $\alpha$. We observe that

[TABLE]

for all $k\geq 0$ for some constant $c_{\alpha}\in\mathbb{Z}$. Indeed, $r$-balls around vertices of distance at least $r$ from one of the corners in $T_{m+k}$ occur exactly $3$-times in $T_{m+k+1}$. In the small set of vertices which lie close to a corner of $T_{m+k}$, we always see two copies of $T_{m}$ being glued at a corner. Which shows that the effect of this operation does not depend on $k$; compare Figure 1.

Now it follows from a short calculation that \Bigl{(}\frac{|N(k,\alpha)|}{|T_{m+k}^{(0)}|}\Bigr{)}_{k} is a Cauchy sequence. Since $r$ and $\alpha$ were arbitrary, we conclude that the sequence $(T_{n})_{n}$ converges in the sense of Benjamini-Schramm.

Now we introduce an action of the finite cyclic group $G=\langle\sigma\rangle$ of order $3$. We let $\sigma$ act by rotation of $2\pi/3$ around the barycenter $c_{n}=2^{n-1}(1,\sqrt{3}^{-1})$ of $T_{n}$. All vertices of $T_{n}$ have Euclidean distance at least $\frac{2^{n-2}}{\sqrt{3}}$ from the barycenter. Thus every vertex is moved by an Euclidean distance of at least $2^{n-2}$ under the non-trivial rotations $\sigma$ and $\sigma^{2}$, which in particular also holds for the path distance in $T_{n}$. Proposition 3.3 implies that the sequence $(T_{n})_{n}$ of simplicial $G$-complexes converges to the induced random rooted simplicial $G$-complex $\operatorname{ind}_{1}^{G}(\tau_{S})$. Roughly speaking, the sequence converges to three copies of the Sierpinski triangle which are permuted cyclically by $G$.

3.1. Towers of finite sheeted covering spaces

In this section we discuss a prominent family of examples of Benjamini-Schramm convergent sequences: towers of finite sheeted covering spaces.

Let $\Gamma$ be a group and let $K$ be simplicial complex of vertex degree at most $D$. Assume that $\Gamma$ acts simplicially on $K$, this means that an element $\gamma\in\Gamma$ stabilizes a simplex of $K$ if and only if it stabilizes all of its vertices. Recall that this condition can always be achieved by passing to the barycentric subdivision of $K$. We assume further that the action is proper and cocompact, i.e., every vertex has a finite stabilizer and there are only finitely many orbits of vertices.

Let $G\leq\Gamma$ be a finite subgroup. For every normal subgroup $N\trianglelefteq\Gamma$ the quotient simplicial complex $K/N$ carries a $G$-action. Suppose that $\Gamma$ is residually finite and let $(N_{n})_{n\in\mathbb{N}}$ be a descending chain of finite index normal subgroups of $\Gamma$ with $\bigcap_{n\in\mathbb{N}}N_{n}=\{1\}$. It is well-known (cf. [13, Example 19]) that the sequence $(K/N_{n})_{n}$ of finite simplicial complexes (without $G$-action) converges to the random rooted simplicial complex

[TABLE]

where $\mathcal{F}$ is a fundamental domain for the action of $\Gamma$ on $K^{(0)}$ and $w(\Gamma)=\sum_{x\in\mathcal{F}}|\operatorname{St}_{\Gamma}(x)|^{-1}$. This measure does not depend on the choice of the fundamental domain. The purpose of this section is to describe the limit taking the $G$-action into account.

We say that an element $\gamma\in\Gamma$ is FC if it has a finite conjugacy class, i.e., $|\Gamma:C_{\Gamma}(\gamma)|<\infty$, where $C_{\Gamma}(\gamma)$ is the centralizer of $\gamma$. Consider the subgroup $H=\{g\in G\mid g\text{ is FC in }\Gamma\}\leq G$ of FC-elements which lie in $G$. Let $\Gamma_{0}\leq_{f.i.}\Gamma$ be a finite index subgroup which satisfies

[TABLE]

Lemma 3.5**.**

Let $\mathcal{F}_{0}\subseteq K^{(0)}$ be a fundamental domain for the action of $\Gamma_{0}$ on $K^{(0)}$. The measure

[TABLE]

on $\mathcal{SC}_{*}^{D}(H)$ is unimodular and does not depend on the choices of $\Gamma_{0}$ and $\mathcal{F}_{0}$.

Proof.

Observe that any element $\gamma\in\Gamma_{0}$ commutes with all $h\in H$ and thus defines an isomorphism between $(K,Hx)$ and $(K,H\gamma x)$ as simplicial $H$-complexes. This shows that the measure is independent of the fundamental domain.

In order to verify that $\mu_{K}^{H}$ does not depend on $\Gamma_{0}$, it is sufficient to show that we can replace $\Gamma_{0}$ by some finite index normal subgroup $\Gamma_{1}\trianglelefteq_{f.i.}\Gamma_{0}$. Let $\mathcal{F}_{1}$ be a fundamental domain for $\Gamma_{1}$. We obtain

[TABLE]

and, from a similar calculation, also $w(\Gamma_{1})=|\Gamma_{0}:\Gamma_{1}|w(\Gamma_{0})$.

It remains to show that $\mu_{K}^{H}$ is unimodular. Let $f\colon\mathcal{SC}_{**}^{D}(H)\to\mathbb{R}_{\geq 0}$ be a measurable function. Unimodularity follows from a short calculation.

[TABLE]

Proposition 3.6**.**

Let $G\leq\Gamma$ be a finite subgroup and let $H\leq G$ be the subgroup of FC-elements for $\Gamma$. Let $(N_{n})_{n}$ be a descending chain of finite index normal subgroups in $\Gamma$ with $\bigcap_{n\in\mathbb{N}}N_{n}=\{1\}$. The sequence of simplicial $G$-complexes $(K/N_{n})_{n}$ converges to the random rooted simplicial complex $\mu_{K}^{G}:=\operatorname{ind}_{H}^{G}(\mu_{K}^{H})$.

Proof.

The proof consists of two steps. First we show that $(K/N_{n})_{n}$ converges as a sequence of $H$-complexes to $\mu_{K}^{H}$ (Claims 1 and 2) and in the second step (Claim 3) we apply Proposition 3.3.

Claim 1: Let $r>0$ be fixed. For all sufficiently large $n$ and all $x\in K^{(0)}$, the $r$-ball $B_{r}(K,Hx)$ in $K$ and the $r$-ball $B_{r}(K/N_{n},HN_{n}x)$ in $K/N_{n}$ are isomorphic as $H$-complexes.

Let $\Gamma_{0}\leq\Gamma$ be as above and let $\mathcal{F}_{0}$ be a fundamental domain for $\Gamma_{0}$ acting on $K^{(0)}$. The action is proper, the sets $\mathcal{F}_{0}$ and $H$ are finite and the vertex degree of $K$ is bounded, hence the set $S$ of elements $\gamma\in\Gamma$ such that

[TABLE]

for some $x_{0}\in\mathcal{F}_{0}$ is finite.

Take $n\in\mathbb{N}$ so large that $S\cap N_{n}=\{1\}$. Then for all $x_{0}\in K^{(0)}$ and $\gamma\in N_{n}$ property (3.2) implies that $\gamma=1$. Indeed, find $\gamma_{0}\in\Gamma_{0}$ with $\gamma_{0}x_{0}\in\mathcal{F}_{0}$ then multiplication with $\gamma_{0}$ yields $B_{r+1}(K,H\gamma_{0}x_{0})\cap\gamma_{0}\gamma\gamma_{0}^{-1}B_{r+1}(K,H\gamma_{0}x_{0})\neq\emptyset$. We deduce $\gamma_{0}\gamma\gamma_{0}^{-1}\in S\cap N_{n}=\{1\}$. In particular, the quotient map takes the vertices of the $r$-ball $B_{r}(K,Hx)$ injectively to the $r$-ball $B_{r}(K/N_{n},HN_{n}x)$. We have to verify that every simplex in $B_{r}(K/N_{n},HN_{n}x)$ lifts to a unique simplex in $B_{r}(K,Hx)$. Let $\sigma$ be a simplex in $B_{r}(K/N_{n},HN_{n}x)$ and let $\tilde{\sigma}$ be a lift in $K$ such that at least one vertex lies in $B_{r}(K,Hx)$. As a consequence $\tilde{\sigma}$ is a simplex in $B_{r+1}(K,Hx)$. Let $y$ be any vertex of $\tilde{\sigma}$. There is an element $k\in N_{n}$ such that $d(ky,Hx)\leq r$. This means that $B_{r+1}(K,Hx)\cap kB_{r+1}(K,Hx)\neq\emptyset$ and shows that $k=1$. In particular, the simplex $\tilde{\sigma}$ lives in $B_{r}(K,Hx)$.

Claim 2: The sequence $\mu_{K/N_{n}}^{H}$ converges to $\mu_{K}^{H}$.

Let $r>0$ and let $\alpha$ be a rooted simplicial $H$-complex of radius at most $r$. Let $n\in\mathbb{N}$ sufficiently large such that $N_{n}$ acts freely on $K$ and so that Claim 1 applies. In addition, we may take $\Gamma_{0}\leq N_{n}$; the action of $\Gamma_{0}$ is also free. Now every point in $K/N_{n}$ is covered by exactly $|N_{n}:\Gamma_{0}|$ points in $\mathcal{F}_{0}$ and we deduce

[TABLE]

Claim 3: (3.1) holds for all $C>0$ and all $g\in G\setminus H$.

Let $i\in\mathbb{N}$ be chosen so that $N_{i}$ acts freely on $K$ and let $\mathcal{F}$ be a fundamental domain for the action of $N_{i}$ on $K^{(0)}$. Let $Z\subseteq\Gamma$ be the finite set of elements $\gamma\in\Gamma$ such that $d(\gamma x,x)\leq C$ for some $x\in\mathcal{F}$. For $n\geq i$ the vertices of $K/N_{n}$ correspond bijectively to $N_{i}/N_{n}\times\mathcal{F}$.

Take $x\in K^{(0)}$ and write $\bar{x}=N_{n}x\in K^{(0)}/N_{n}$. Suppose that $\bar{x}\in E(K/N_{n},g,C)$; i.e., there is $\gamma_{n}\in N_{n}$ with $d(gx,\gamma_{n}x)\leq C$. There is a unique $x_{0}\in\mathcal{F}$ and an element $\gamma_{i}\in N_{i}$ satisfying $x=\gamma_{i}x_{0}$. This shows that $d(\gamma_{i}^{-1}\gamma_{n}^{-1}g\gamma_{i}x_{0},x_{0})\leq C$ and so $\gamma_{i}^{-1}g\gamma_{i}\in ZN_{n}$. How many elements has the finite set $e_{n}(g,Z)=\{k\in N_{i}/N_{n}\mid k^{-1}gk\in ZN_{n}/N_{n}\}$? Clearly, its cardinality is bounded above by $|Z|\cdot|C_{N_{i}/N_{n}}(gN_{n})|$. The element $g\in G$ has an infinite conjugacy class in $\Gamma$ and thus

[TABLE]

see the proof of [10, Lemma 4.12]. We deduce that

[TABLE]

This proves the last claim and Proposition 3.3 completes the proof. ∎

4. Homology and $\ell^{2}$-multiplicities of random rooted complexes

4.1. The homology of a random rooted complex

In order to define $\ell^{2}$-multiplicities we introduce the $\ell^{2}$-homology of random rooted simplicial $G$-complexes. To this end, we will construct a chain complex for each probability measure on $\mathcal{SC}_{*}^{D}(G)$. We begin with a technical ingredient which allows us to pick a representative for each isomorphism class $[K,o]\in\mathcal{SC}_{*}^{D}(G)$ of rooted simplicial $G$-complexes in a measurable way.

Let

[TABLE]

and let $\Delta^{D}(\mathbb{N}_{G})$ be the simplicial $G$-complex consisting of all finite nonempty subsets of $\mathbb{N}_{G}$ with at most $D+1$ elements. The action of $G$ is defined via the second coordinate. Every subcomplex $S$ of $\Delta^{D}(\mathbb{N}_{G})$ can be encoded by an element $f_{S}\in\{0,1\}^{\Delta^{D}(\mathbb{N}_{G})}$ such that $f_{S}(\sigma)=1$ exactly if the simplex $\sigma$ is contained in the subcomplex $S$. We endow $\{0,1\}^{\Delta^{D}(\mathbb{N}_{G})}$ with the product topology, i.e., the topology generated by all cylinder sets. Let $\mathrm{Sub}(\Delta^{D}(\mathbb{N}_{G}))\subseteq\{0,1\}^{\Delta^{D}(\mathbb{N}_{G})}$ be the subset which consists of elements encoding $G$-invariant subcomplexes which contain a unique orbit of the form $\{0\}\times X$; this is a closed subspace.

Lemma 4.1**.**

There is a continuous map $\Psi\colon\mathcal{SC}_{*}^{D}(G)\to\mathrm{Sub}(\Delta^{D}(\mathbb{N}_{G}))$ such that $(\Psi([K,o]),\{0\}\times X)$ is a representative of $[K,o]$ for all $[K,o]\in\mathcal{SC}_{*}^{D}(G)$.

Proof.

We only sketch the proof; a detailed treatment of the nonequivariant case can be found in [13, Lemma 1].

We enumerate $\mathbb{N}_{G}$, the set of vertices of $\Delta^{D}(\mathbb{N}_{G})$, in the following way. First we enumerate the (isomorphism classes of) $G$-sets $X_{1},\dots,X_{k}\in\operatorname{Orb}(G)$ and for every $i$ we order the elements of $X_{i}=\{x_{i,1},...,x_{i,m_{i}}\}$. Finally, we enumerate $\mathbb{N}_{G}$ diagonally:

[TABLE]

Once the set of vertices is ordered, a diagonal enumeration provides an order on the set of all simplices of $\Delta^{D}(\mathbb{N}_{G})$. The lexicographic order on $\{0,1\}^{\Delta^{D}(\mathbb{N}_{G})}$, given by $a\prec b$ if there is a simplex $\sigma_{0}$ such that $a(\sigma)=b(\sigma)$ for all $\sigma<\sigma_{0}$ and $a(\sigma_{0})=1$ but $b(\sigma_{0})=0$, defines an order on the subcomplexes of $\Delta^{D}(\mathbb{N}_{G})$. We define $\Psi$ to map an isomorphism class $[K,o]\in\mathcal{SC}_{*}^{D}(G)$ to the $\prec$-minimal subcomplex $\Lambda$ of $\Delta^{D}(\mathbb{N}_{G})$ such that $(\Lambda,\{0\}\times X_{i})\in[K,o]$, where $X_{i}$ is the $G$-set in $\operatorname{Orb}(G)$ with $X_{i}\cong o$, and such that the elements of $\{0\}\times X_{i}$ are the only vertices of $\Lambda$ with first coordinate [math]. For a finite simplicial $G$-complex the existence of the minimal subcomplex of $\Delta^{D}(\mathbb{N}_{G})$ follows from the well-ordering principle. In the situation of an infinite simplicial $G$-complex $[K,o]$ it is a direct consequence of the fact that

[TABLE]

The preimage of a cylinder set under $\Psi$ is a countable union of open sets $U(\alpha,r)$ in $\mathcal{SC}_{*}^{D}(G)$ and therefore open, hence $\Psi$ is continuous. ∎

For any simplicial complex $L$, we write $C^{(2)}_{n}(L)$ to denote the complex Hilbert space of square-summable oriented $n$-chains of $L$. The map $\Psi$ from the preceding lemma gives rise to a field of Hilbert spaces $[K,o]\to C^{(2)}_{n}(\Psi([K,o]))$ on $\mathcal{SC}_{*}^{D}(G)$ for each $n\in\mathbb{N}$. In addition, every oriented $n$-simplex $s$ of $\Delta^{D}(\mathbb{N}_{G})$ yields a characteristic vector field $\xi_{s}$ defined as

[TABLE]

We observe that, since $\Psi$ is continuous, the function $[K,o]\mapsto\langle\delta_{s}([K,o]),\delta_{s^{\prime}}([K,o])\rangle$ is continuous for all oriented simplices $s$ and $s^{\prime}$; the $\xi_{s}$ form a fundamental sequence for a measurable field of Hilbert spaces; see [6, Prop. 4]. A vector field $\sigma\colon[K,o]\mapsto\sigma([K,o])\in C^{(2)}_{n}(\Psi([K,o]))$ is called measurable, if

[TABLE]

is measurable for every oriented $n$-simplex $s$ of $\Delta^{D}(\mathbb{N}_{G})$. Let $\mu$ be a random rooted simplicial $G$-complex. The measurable vector fields $\sigma$ with the property

[TABLE]

form a pre-Hilbert space using the inner product

[TABLE]

By factoring out the subspace of vector fields which vanish almost everywhere, we obtain a Hilbert space: the associated direct integral;

[TABLE]

for details see [6, p. 168]. The differentials $\partial_{n,[K,o]}$ and their adjoints $d_{[K,o]}^{n}$ of the fibres $C^{(2)}_{n}(\Psi([K,o])$ define bounded operators (compare to [13])

[TABLE]

which commute with the induced unitary $G$-action on $C^{(2)}_{n}(\mathcal{SC}_{*}^{D}(G),\mu)$, since they commute fibrewise and $G$ preserves fibres. Therefore, we have for each random rooted simplicial $G$-complex a chain complex $C^{(2)}_{*}(\mathcal{SC}_{*}^{D}(G),\mu)$ and a Laplace operator $\Delta_{n}=\partial_{n+1}\circ d_{n+1}+d_{n}\circ\partial_{n}$ which also commutes with the $G$-action.

Definition 4.2**.**

We define the $n$-th simplicial $\ell^{2}$-homology of a random rooted simplicial complex $\mu$ as the Hilbert space

[TABLE]

equipped with the natural unitary action of $G$.

We would like to have a notion of dimension for a subspace of $C^{(2)}_{n}(\mathcal{SC}_{*}^{D}(G),\mu)$, to this end we introduce a von Neumann algebra with a trace. A bounded linear operator $T$ on $C^{(2)}_{n}(\mathcal{SC}_{*}^{D}(G),\mu)$ is decomposable, if there is an essentially bounded measurable field of operators $[K,o]\mapsto T_{[K,o]}$ such that $T=\int^{\oplus}T_{[K,o]}d\mu([K,o])$; see [6, p. 182]. The bounded decomposable operators $T$ on $C^{(2)}_{n}(\mathcal{SC}_{*}^{D}(G),\mu)$ such that for almost all $[K,o]$ and for all isomorphisms $\varphi\colon\Psi([K,o])\to\Psi([K,o^{\prime}])$ of simplicial $G$-complexes the identity

[TABLE]

holds, form a von Neumann algebra $\mathcal{A}_{n}(\mu)$. In fact, to see that $\mathcal{A}_{n}(\mu)$ is closed in the strong operator topology one can use [6, Prop. 4, p. 183]. Of course, the operators defined by elements of $G$ are contained in $\mathcal{A}_{n}(\mu)$, since we only consider isomorphisms which commute with the $G$-action. Moreover, $\Delta_{n}\in\mathcal{A}_{n}(\mu)$, because $\partial_{*}$ and $d_{*}$ commute with the chain map $\varphi_{\sharp}\colon C^{(2)}_{n}(\Psi([K,o]))\to C^{(2)}_{n}(\Psi([K,o^{\prime}]))$ induced by an isomorphism. For $T\in\mathcal{A}_{n}(\mu)$ we define

[TABLE]

where $\Delta^{D}(\mathbb{N}_{G})(n)$ denotes the set of $n$-simplices of $\Delta^{D}(\mathbb{N}_{G})$. Note that the formula does not depend on the chosen orientation of $s$. As in the nonequivariant case one can verify that $\operatorname{tr}(ST)=\operatorname{tr}(TS)$; see [13] after Definition 5. We obtain a normal, faithful and finite trace on $\mathcal{A}_{n}(\mu)$; compare to [13, Prop. 3].

Definition 4.3**.**

Let $\mathcal{K}$ be a field of $G$-invariant subspaces of $C_{n}^{(2)}(\mathcal{SC}_{*}^{D}(G),\mu)$ such that $\varphi_{\sharp}\mathcal{K}([K,o])=\mathcal{K}([K,o^{\prime}])$ for every isomorphism $\varphi\colon\Psi([K,o])\to\Psi([K,o^{\prime}])$. Then the projection $\operatorname{pr}_{\mathcal{K}}\colon[K,o]\mapsto\operatorname{pr}_{\mathcal{K}([K,o])}$ is an element of $\mathcal{A}_{n}(\mu)$ and we define the von Neumann dimension of $\mathcal{K}$ as

[TABLE]

Example 4.4**.**

Let $L$ be a finite simplicial $G$-complex and $\mu_{L}^{G}$ the associated random rooted simplicial $G$-complex from Example 2.3. Let $\mathcal{K}$ be a field of $G$-invariant subspaces of $C_{n}^{(2)}(\mathcal{SC}_{*}^{D}(G),\mu_{L}^{G})$ as in Definition 4.3. Given an orbit $o\subseteq L^{(0)}$ and an isomorphism $\eta\colon L\to\Psi([L,o])$ we define $\mathcal{K}(L)=\eta^{-1}(\mathcal{K}([L,o]))\subseteq C_{n}^{(2)}(L)$; this subspace does not depend on $o$ and $\eta$. We compute the dimension:

[TABLE]

4.2. Approximation of $\ell^{2}$-multiplicities

In order to fix our notation we recall some facts from the representation theory of finite groups. Let $G$ be a finite group and let $(\rho,V)\in\operatorname{Irr}(G)$ be an irreducible representation of $G$ on the complex vector space $V$. Usually we will not mention the underlying vector space and simply speak of the representation $\rho$. Every irreducible representation is finite dimensional and is uniquely determined by its character $\chi_{\rho}\colon G\to\mathbb{C}$ which maps $g\in G$ to the trace of $\rho(g)$. In particular, $\chi_{\rho}(1)$ is the dimension of the underlying space $V$.

Let $(\sigma,W)$ be any finite dimensional complex representation of $G$, then $W$ can be decomposed into isotypic components

[TABLE]

where each $W_{\rho}$ is (noncanonically) isomorphic to a direct sum of copies of $\rho$, i.e., $W_{\rho}\cong\rho^{m(\rho,\sigma)}$ where the number $m(\rho,\sigma)$ of copies is called the multiplicity of $\rho$ in $\sigma$.

The irreducible representations correspond bijectively to the central idempotents in the group ring $\mathbb{C}[G]$. The central idempotent corresponding to $\rho$ is

[TABLE]

see [9, (2.12)]. The element $P_{\rho}$ defines the orthogonal projection onto the $\rho$-isotypic component in every unitary representation of $G$.

Definition 4.5**.**

Let $\mu$ be a random rooted simplicial $G$-complex and $\rho$ an irreducible representation of $G$. The $\ell^{2}$-multiplicity of $\rho$ in the homology of $\mu$ is

[TABLE]

where $H^{(2)}(\mathcal{SC}_{*}^{D}(G),\mu)_{\rho}$ denotes the $\rho$-isotypic component in the homology of $\mu$.

In addition, we define the $n$-th $\rho$-Laplacian to be

[TABLE]

where $\Delta_{n}$ is the Laplacian of $C^{(2)}_{n}(\mathcal{SC}_{*}^{D}(G),\mu)$.

Remark 4.6**.**

If $G$ is the trivial group $\{1\}$ and $\rho$ is the unique irreducible representation of $G$, i.e., the trivial $1$-dimensional representation, then $m^{(2)}_{n}(\rho,\mu)=b_{n}^{(2)}(\mu)$ is simply the $n$-th $\ell^{2}$-Betti number of a random rooted simplicial complex defined in [13, Def. 6].

Example 4.7**.**

Let $K$ be a finite simplicial $G$-complex and $\mu_{K}^{G}$ the associated random rooted simplicial $G$-complex. Then it follows from Example 4.4 that $m_{n}^{(2)}(\rho,\mu_{K}^{G})$ is the ordinary multiplicity of the representation $\rho$ in $H_{n}(K,\mathbb{C})$ divided by the number of vertices, i.e.,

[TABLE]

Lemma 4.8**.**

The operator $\Delta_{n,\rho}$ is positive self-adjoint and the operator norm $\|\Delta_{n,\rho}\|$ is bounded above by a constant $b(n,D)$ which depends only on $n$ and $D$. Moreover, the kernel of $\Delta_{n,\rho}$ is $H^{(2)}_{n}(\mathcal{SC}_{*}^{D}(G),\mu)_{\rho}$.

Proof.

It is easy to see that the operator $\Delta_{n,\rho}$ is positive self-adjoint using that $\Delta_{n}$ and $\operatorname{Id}-P_{\rho}$ have these properties and commute. It is well-known (see [13, Proposition 2]) that the operator norm of the Laplacian $\Delta_{n}$ is bounded, since the bound on the vertex degree yields a bound for the number of $(n+1)$-simplices which contain a given $n$-simplex. Now, the operator $\operatorname{Id}-P_{\rho}$ is a projection and we find $\|\Delta_{n,\rho}\|\leq\|\Delta_{n}\|+1$.

Observe that a vector $x$ lies in $\ker(\Delta_{n,\rho})$ if and only if $\langle\Delta_{n,\rho}x,\Delta_{n,\rho}x\rangle=0$. We note further that

[TABLE]

since $P_{\rho}$ and $\Delta_{n}$ are self-adjoint and commute. All three summands are nonnegative, since $\Delta_{n}(\operatorname{Id}-P_{\rho})$ is a positive operator. We conclude that

[TABLE]

∎

Let $E_{\Delta_{n,\rho}}$ be the unique projection valued measure obtained from the spectral calculus for the bounded and self-adjoint operator $\Delta_{n,\rho}$. Then $E_{\Delta_{n,\rho}}$ has the property that for all bounded Borel functions $f$ on $\mathbb{R}$

[TABLE]

Further, we define the *spectral measure of * $\Delta_{n,\rho}$ as

[TABLE]

for every Borel set $B$. The spectral measure satisfies $\operatorname{tr}(f(\Delta_{n,\rho}))=\int_{\mathbb{R}}f(\lambda)d\nu_{n,\rho}$ for all bounded Borel functions $f$ on $\mathbb{R}$.

Lemma 4.9**.**

Let $(\mu^{k})_{k\in\mathbb{N}}$ be a sequence of random rooted simplicial $G$-complexes which converges weakly to $\mu^{\infty}$ and let $\nu^{k}_{n,\rho}$ be the associated spectral measures of the $n$-th $\rho$-Laplacians $\Delta_{n,\rho}$. Then $(\nu^{k}_{n,\rho})_{k}$ converges weakly to $\nu^{\infty}_{n,\rho}$.

Proof.

For the sake of simplicity we denote $\nu^{k}_{n,\rho}$ and $\nu^{\infty}_{n,\rho}$ by $\nu^{k}$ and $\nu^{\infty}$ respectively. Since, by Lemma 4.8, $\operatorname{spec}(\Delta_{n,\rho})\subseteq[0,R]$ for some $R>0$, Weierstraß approximation implies that it is enough to check the identity

[TABLE]

for all polynomials $f\in\mathbb{R}[x]$. By linearity we can further assume that $f=x^{r}$.

[TABLE]

Let us consider $\Delta_{n,\rho}^{r}$ and observe that

[TABLE]

Let $s\in\Delta(\mathbb{N}_{G})(n)$ with $(0,x)\in s$ and $x\in X$ and suppose that $s\in\Psi([K,o])$ for a rooted isomorphism class $[K,o]$. Then $(\operatorname{Id}-P_{\rho})(s)$ is supported in the $1$-ball of the orbit $\{0\}\times X$, since $P_{\rho}$ is a linear combination of elements $g\in G$ which only act on the second coordinate. Further, $\Delta_{n}(s)$ is a linear combination of simplices in the $2$-ball around $(0,x)$. Therefore, $\langle\Delta_{n,\rho}^{r}\xi_{s},\xi_{s}\rangle$ only depends on the $2r+1$-neighbourhood of the orbit $\{0\}\times X$. By the weak convergence of the sequence $(\mu^{k})_{k}$ we obtain that

[TABLE]

for all finite rooted simplicial $G$-complexes $\alpha$. Now the claim follows from the fact that $\mathcal{SC}_{*}^{D}(G)$ is a finite union of open sets of the form $U(\alpha,2r+1)$. ∎

Now we can prove Theorem 1.1. We follow the well-known strategy, going back to Lück [11] and Schick [12], of bounding the Fuglede-Kadison determinant. Here we have to use nonrational algebraic coefficients and the corresponding method is inspired from [7, Section 3]; compare also [10, Lemma 3.14].

Proof of Theorem 1.1.

Since all $\mu_{k}$ are sofic, it is sufficient to prove the theorem under the assumption that the sequence $\mu_{k}=\mu_{K_{k}}^{G}$ is actually a sequence of finite simplical $G$-complexes. Let $\nu^{k}_{n,\rho}$ (resp. $\nu^{\infty}_{n,\rho}$) be the spectral measure of the $n$-th $\rho$-Laplacian of $K_{k}$ (resp. of $\mu_{\infty}$).

By Lemma 4.8 we have to show that $\nu_{n,\rho}^{k}(\{0\})=\chi_{\rho}(1)m_{n}^{(2)}(\rho,\mu_{K_{k}}^{G})$ converges to $\nu_{n,\rho}^{\infty}(\{0\})$. In view of Lemma 4.9 it remains to show that the Fuglede-Kadison determinant

[TABLE]

of $\nu^{k}_{n,\rho}$ is uniformly bounded away from zero; see [10, Lemma 2.20].

Let $E\subseteq\mathbb{C}$ be a finite Galois extension of $\mathbb{Q}$ which is a splitting field for the finite group $G$; see [9, (9.10)] for the existence. In particular, all irreducible characters of $G$ take values only in the ring of integers $\mathcal{O}_{E}$ of $E$; see [9, (3.6)].

Fix $k\in\mathbb{N}$. We pick a basis of $C_{n}^{(2)}(K_{k})$ by choosing an orientation for every $n$-simplex of $K=K_{k}$. The transformation matrix $A_{\rho}$ of the $\rho$-Laplacian $\Delta_{n,\rho}$ on $K_{k}$ with respect to this basis has entries in $\frac{1}{|G|}\mathcal{O}_{E}$. The spectral measure $\nu^{k}_{n,\rho}$ agrees with the spectral measure of $A_{\rho}$ normalized by the number of vertices $|K^{(0)}|$; cf. Example 4.4. In particular, the power $\det(\nu^{k}_{n,\rho})^{|K^{(0)}|}$ of the Fuglede-Kadison determinant is just the product over all non-zero eigenvalues of $A_{\rho}$, i.e., the lowest non-zero coefficient $c_{\rho}$ of the characteristic polynomial of $A_{\rho}$. We note that $c_{\rho}\in|G|^{-|K(n)|}\mathcal{O}_{E}$ and since, by Lemma 4.8, the operator norm of $\Delta_{n,\rho}$ is bounded above, there is an upper bound $|c_{\rho}|\leq b^{|K(n)|}$ where $b$ depends only on $n$ and $D$.

Consider the action of the Galois group $\mathrm{Gal}(E/\mathbb{Q})$ on the irreducible representations of $G$; cf. [9, p.152]. If $\tau\in\mathrm{Gal}(E/\mathbb{Q})$ be a Galois automorphism of $E$, then $\tau(A_{\rho})=A_{\tau(\rho)}$ and $\tau(c_{\rho})=c_{\tau(\rho)}$. In particular, we obtain

[TABLE]

and therefore $|c_{\rho}|\geq|G|^{-[E:\mathbb{Q}]|K(n)|}b^{-([E:\mathbb{Q}]-1)|K(n)|}$. Since the vertex degree is bounded above by $D$, there is a constant $t>0$ only depending on $n$ and $D$ such that $|K(n)|\leq t|K^{(0)}|$. We conclude that

[TABLE]

and this lower bound does not depend on $K_{k}$. ∎

Let $G$ be a finite group and $H$ be a subgroup. Let $(\sigma,V)$ be a finite dimensional complex representation of $H$. Recall that the multiplicity of an irreducible representation $\rho\in\operatorname{Irr}(G)$ in $\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V$ is determined by the Forbenius receprocity formula

[TABLE]

see [9, (5.2)]. The following theorem provides a reciprocity formula for induced sofic random rooted $G$-complexes. In particular, with $H=\{1\}$ we obtain Theorem 1.2 stated in the introduction.

Theorem 4.10**.**

Let $\mu$ be a random rooted simplicial $H$-complex. If $\mu$ is sofic, then $\operatorname{ind}_{H}^{G}(\mu)$ is sofic and, moreover,

[TABLE]

for every $\rho\in\operatorname{Irr}(G)$ and all $n\in\mathbb{N}_{0}$.

Proof.

We write $\nu=\operatorname{ind}_{H}^{G}(\mu)$. Since $\mu$ is sofic, we can find a sequence $K_{k}$ of finite simplicial $H$-complexes such that the associated random rooted simplicial complexes $\mu_{k}$ converge weakly to $\mu$. Continuity of induction (Lemma 3.1) implies that the sequence $\operatorname{ind}_{H}^{G}(\mu_{k})$ converges to $\nu$. This shows that $\nu$ is sofic, since $\operatorname{ind}_{H}^{G}(\mu_{k})$ is the random rooted simplicial $G$-complex defined by $W_{k}=G\times_{H}K_{k}$.

Finally, using that $W_{k}$ is a disjoint union of $|G/H|$ copies of $K_{k}$ which are permuted by the action it follows that

[TABLE]

In particular, Frobenius reciprocity implies that

[TABLE]

Based on the relation of multiplicities and $\ell^{2}$-multiplicities from Example 4.7 an application of Theorem 1.1 completes the proof. ∎

Remark 4.11**.**

It appears to us that formula (4.1) should hold without the assumption of soficity. However, due to some technical problems and based on the fact that we do not know a single example of a non-sofic random rooted simplicial complex, we decided to restrict to the sofic case.

Example 4.12**.**

(The $\ell^{2}$-multiplicities of Sierpinski’s triangle) We return to the setting of Example 3.4 and we compute the $\ell^{2}$-mulitplicities of Sierpinski’s triangle with the rotation action of the cyclic group $G=\langle\sigma\rangle$ of order $3$. Recall that, with this action, the Sierpinski triangle is an induced random rooted simplicial complex $\operatorname{ind}_{1}^{G}(\tau_{S})$ where $\tau_{S}$ is the limit of a sequence of finite $2$-dimensional simplicial complexes $T_{n}$. Therefore, by Theorem 1.2 it is sufficient to compute the $\ell^{2}$-Betti numbers of $\tau_{S}$.

In order to use the Approximation Theorem 1.1 for the action of the trivial group (compare also [13]), we compute the normalized Betti numbers of every $T_{n}$. Note that $T_{n}$ is homotopy equivalent to a $1$-dimensional complex, thus it is sufficient to calculate the normalized Euler characteristic of $T_{n}$. Using the formulas given in Example 3.4 we find

[TABLE]

Every $T_{n}$ is connected, so $b_{0}(T_{n})=1$ for all $n$ and the normalized zeroth Betti numbers tend to zero; i.e., $b_{0}^{(2)}(\tau_{S})=0$. We deduce that $b_{1}^{(2)}(\tau_{S})=\frac{1}{3}$. Finally, we apply Theorem 1.2 to deduce that $m^{(2)}_{1}(\rho,\operatorname{ind}_{1}^{G}(\tau_{S}))=\frac{1}{9}$ for every irreducible representation $\rho\in\operatorname{Irr}(G)$. All $\ell^{2}$-multiplicities of Sierpinski’s triangle vanish outside of degree $1$.