Let G be a finitely presented group, and let {G_i} be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1. G_i is an amalgamated free product or HNN extension, for infinitely many i; 2. the Cayley graphs of G/G_i (with respect to a fixed finite set of generators for G) form an expanding family; 3. inf_i (d(G_i)-1)/[G:G_i] = 0, where d(G_i) is the rank of G_i.
The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.
Using the construction of a nonorientable Curtis-Tits group of type $\tilde A_n$, we obtain new explicit families of expander graphs of valency five for unitary groups over finite fields.
We study the representations of non-commutative universal lattices and use them to compute lower bounds for the \TauC for the commutative universal lattices $G_{d,k}= \SL_d(\Z[x_1,...,x_k])$ with respect to several generating sets. As an application of the above result we show that the Cayley graphs of the finite groups $\SL_{3k}(\F_p)$ can be made expanders using suitable choice of the generators. This provides the first examples of expander families of groups of Lie type where the rank is not bounded and gives a natural (and explicit) counter examples to two conjectures of Alex Lubotzky and …
We introduce a generalization of expander graphs, which is called a weak expander sequence. It is proved that a uniform Roe algebra of a weak expander sequence is not locally reflexive. It follows that uniform Roe algebras of expander graphs are not exact. We introduce the notion of a generalized box space to discuss box spaces and expander sequences in a unified framework. Key tools for the proof are amenable traces and measured groupoids associated to generalized box spaces.
We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling and other areas.
This article contains two rigidity type results for $\mathrm{SL}(n,\mathbb{Z})$ for large $n$ that share the same proof. Firstly, we prove that for every $p \in [1,\infty]$ different from $2$, the noncommutative $L^p$-space associated with $\mathrm{SL}(n,\mathbb{Z})$ does not have the completely bounded approximation property for sufficiently large $n$ depending on $p$. The second result concerns the coarse embeddability of expander families constructed from $\mathrm{SL}(n,\mathbb{Z})$. Let $X$ be a Banach space and suppose that there exist $β< \frac{1}{2}$ and $C>0$ such that the Banach-Mazur…
In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space $X$. Expanders with this girth property are a necessary ingredient in the construction of the so-called `Gromov monster' groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum-Connes assembly map with certain coefficients is injectiv…
We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equals 2) and our second family is given as Cayley graphs of finite groups with very short presentations with only 2 generators and 4 relations. Both families are based on particular finite quotients of a group G of infinite upper triangular matrices over the ring M(3,F2). We present explicit vector space bases for the finite abelian quotients of the lower exponent-2 groups of G by upper tri…
A regular equivalence between two graphs $Γ,Γ'$ is a pair of uniformly proper Lipschitz maps $VΓ\to VΓ'$ and $VΓ'\to VΓ$. Using separation profiles we prove that there are $2^{\aleph_0}$ regular equivalence classes of expander graphs, and of finitely generated groups with a representative which isometrically contains expanders.
The purpose of this article is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs $\{Γ_i\}_{i\in\mathbb{N}}$ forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups $\{A(Γ_i)\}_{i\in\mathbb{N}}$ agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of \emph{vector space expanders}, a novel struc…
Let $Φ$ be an irreducible root system (other than $G_2$) of rank at least $2$, let $\mathbb{F}$ be a finite field with $p = \operatorname{char} \mathbb{F} > 3$, and let $\mathrm{G}(Φ,\mathbb{F})$ be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension $\mathrm{rank}(Φ)$, where $\mathrm{G}(Φ,\mathbb{F})$ acts simply transitively on the top-dimensional faces; these are $λ$-spectral HDXs with $λ\to 0$ as $p \to \infty$. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case $Φ= A_d$. Our…
For n at least 2, the concept of n-way expanders was defined by various researchers. Bigger n gives a weaker notion in general, and 2-way expanders coincide with expanders in usual sense. Koji Fujiwara asked whether these concepts are equivalent to that of ordinary expanders for all n for a sequence of Cayley graphs. In this paper, we answer his question in the affirmative. Furthermore, we obtain universal inequalities on multi-way isoperimetric constants on any finite connected vertex-transitive graph, and show that gaps between these constants imply the imprimitivity of the group action on t…
Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes.
Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways (cf. [Lub94], [HLW06], [Lub12] and the references therein). In the last decade, a theory of "high dimensional expanders" has begun to emerge. The goal of the current paper is to describe some paths of this new area of study.
In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum-Connes assembly map is an isomorphism for the associated metric space $X$. As discussed in the first paper in this series, this has applications to the Baum-Connes conjecture for `Gromov monster' groups. We also introduce a new property, `geometric property (T)'. For the metric space associated to a sequence of graphs, this property is an obstruction to the maximal coarse assembly map being an…
Let the group $G$ act transitively on the finite set $Ω$, and let $S \subseteq G$ be closed under taking inverses. The Schreier graph $Sch(G \circlearrowleft Ω,S)$ is the graph with vertex set $Ω$ and edge set $\{ (ω,ω^s) : ω\in Ω, s \in S \}$. In this paper, we show that random Schreier graphs on $C \log|Ω|$ elements exhibit a (two-sided) spectral gap with high probability, magnifying a well known theorem of Alon and Roichman for Cayley graphs. On the other hand, depending on the particular action of $G$ on $Ω$, we give a lower bound on the number of elements which are necessary to provide a …
We show that pairs of generators for the family Sz(q) of Suzuki groups may be selected so that the corresponding Cayley graphs are expanders. By combining this with several deep works of Kassabov, Lubotzky and Nikolov, this establishes that the family of all non-abelian finite simple groups can be made into expanders in a uniform fashion.
We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong exp…
We prove that there exist $k\in N$ and $0<ε\in R$ such that every non-abelian finite simple group $G$, which is not a Suzuki group, has a set of $k$ generators for which the Cayley graph $\Cay(G; S)$ is an $ε$-expander.
We construct an explicit generating sets $F_n$ and $\tilde F_n$ of the alternating and the symmetric groups, which make the Cayley graphs $C(Alt(n), F_n)$ and $C(Sym(n), \tilde F_n)$ a family of bounded degree expanders for all sufficiently large $n$. These expanders have many applications in the theory of random walks on groups and other areas of mathematics.
For a Banach space $X$, we show that any family of graphs quasi-isometric to levels of a warped cone $\mathcal O_ΓY$ is an expander with respect to $X$ if and only if the induced $Γ$-representation on $L^2(Y;X)$ has a spectral gap. This provides examples of graphs that are an expander with respect to all Banach spaces of non-trivial type.
A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their ge…
We provide an explicit construction of finite 4-regular graphs $(Γ_k)_{k\in \mathbb N}$ with ${girth Γ_k\to\infty}$ as $k\to\infty$ and $\frac{diam Γ_k}{girth Γ_k}\leqslant D$ for some $D>0$ and all $k\in\mathbb{N}$. For each fixed dimension $n\geqslant 2,$ we find a pair of matrices in $SL_{n}(\mathbb{Z})$ such that (i) they generate a free subgroup, (ii)~their reductions $\bmod\, p$ generate $SL_{n}(\mathbb{F}_{p})$ for all sufficiently large primes $p$, (iii) the corresponding Cayley graphs of $SL_{n}(\mathbb{F}_{p})$ have girth at least $c_n\log p$ for some $c_n>0$. Relying on growth resul…
