Related papers: On sofic approximations of Property (T) groups
We construct p.m.p. group actions that are not local-global limits of sequences of finite graphs. Moreover, they do not weakly contain any sequence of finite labeled graphs. Our methods are based on the study of almost automorphisms of…
D. A. Kahzdan first put forth property (T) in relation to the study of discrete subgroups of Lie groups of finite co-volume. Through a combinatorial approach, we define an analogue of property (T) for regular graphs. We then prove the basic…
We prove that every sofic approximation of a property (T) group is approximately isomorphic to one having geometric property (T), and more generally, a box space of graphs which has boundary geometric property (T) is approximately…
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group…
It is shown that there are groups $\Gamma$ with finite generating sets $S$ such that the adjacency operator of the Cayley graph ${\rm Cay}(\Gamma,S)$ is a disjoint union of $N$ intervals, for arbitrarily large integers $N$.
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…
A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…
Given a finitely generated group $G$ that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of $G$ is…
An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product…
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…
We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings'…
Applying a well-known theorem due to Eidelheit, we give a short proof of the surjectivity of the combinatorial Laplacian on connected locally finite undirected simplicial graph $G$ with countably infinite vertex set $V$, established by…
We prove that uniform Roe C*-algebras associated to some expander graphs coming from discrete groups with property (\tau) are not K-exact. In particular, we show that this is the case for the expander obtained as Cayley graphs of a sequence…
For a transitive infinite connected graph $G$, let $\mu(G)$ be its connective constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of…
The objective of this series is to study metric geometric properties of disjoint unions of amenable Cayley graphs by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint…
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper, establishing strongly…
We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In…
Let $\mathbf G$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf G$ is {\em treeable} then it arises as the local limit of some sequence $(G_n)_{n\in\mathbb{N}}$ of graphs with…
In this series of papers, we study correspondence between the following: (1) large scale structure of the metric space bigsqcup_m {Cay(G(m))} consisting of Cayley graphs of finite groups with k generators; (2) structure of groups which…