Related papers: Unimodular random graphs with Property (T) have co…

George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values…

Given an edge-independent random graph G(n,p), we determine various facts about the cohomology of graph products of groups for the graph G(n,p). In particular, the random graph product of a sequence of finite groups is a rational duality…

Erd\H{o}s and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ (a graph is $H$-free if it has no induced subgraph isomorphic to $H$).…

This paper investigates when countable graphs have a finite or an infinite chromatic number through model theoretic methods. For Fra\"{i}ss\'{e} limits, we show that instability forces the chromatic number to be infinite, yielding a…

We prove that an arbitrary compact metrizable group can be realized as the automorphism group of a graphing; this is a continuous analogue to Frucht's theorem recovering arbitrary finite groups are automorphism groups of finite graphs. The…

The $k$-gonal models of random groups are defined as the quotients of free groups on $n$ generators by cyclically reduced words of length $k$. As $k$ tends to infinity, this model approaches the Gromov density model. In this paper we show…

Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth…

A random rooted graph is said to be sofic if it is the Benjamini-Schramm limit of a sequence of finite graphs. Given any finite graph $H$, we prove that every one-ended, unimodular random rooted graph that does not have H as a minor must be…

Let P_{n,d,D} denote the graph taken uniformly at random from the set of all labelled planar graphs on {1,2,...,n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability…

Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container…

If $N \subset \R$ is a separable II$_1$-factor, the space $\Hom(N,\R)$ of unitary equivalence classes of unital *-homomorphisms $N \to \R$ is shown to have a surprisingly rich structure. If $N$ is not hyperfinite, $\Hom(N,\R)$ is an…

We show that the main results of Happel-Rickard-Schofield (1988) and Happel-Reiten-Smalo (1996) on piecewise hereditary algebras are coherent with the notion of group action on an algebra. Then, we take advantage of this compatibility and…

If P is a hereditary property then we show that, for the existence of a perfect f-factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size aleph_1. Further we give two examples of a…

We study the problem HomsTo$H$ of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph $H$. A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than…

We characterize which groups splitting as finite graphs of free groups with cyclic edge groups are residually finite. Such a group $G$ is residually finite if and only if all its Baumslag-Solitar subgroups are residually finite. From a…

We propose to study a natural version of Connes' Rigidity Conjecture that involves property (T) groups with infinite center. Utilizing techniques at the intersection of von Neumann algebras and geometric group theory, we establish several…

We expose a class of discrete metric spaces, for which bounded geometry is equivalent to the property A of G. Yu. This class includes the coarse disjoint union of $(\mathbb Z/2\mathbb Z)^n$, $n\in\mathbb N$, and consists of spaces of simple…

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to…

We prove Bowen's conjecture that every sequence of finite graphs that locally converges to the Cayley graph of a countably infinite group with Kazhdan Property (T) is essentially a vertex-disjoint union of expander graphs. We characterize…

We show that if $G$ is a non-archimedean, Roelcke precompact, Polish group, then $G$ has Kazhdan's property (T). Moreover, if $G$ has a smallest open subgroup of finite index, then $G$ has a finite Kazhdan set. Examples of such $G$ include…