Related papers: Random groups at density $d<3/14$ act non-triviall…

We provide geometric methods to give bounds on the large-scale dimension of CAT(0) cube complexes quasiisometric to a given group $G$. In situations where these bounds conflict we obtain obstructions to $G$ being cocompactly cubulated. More…

We prove a compactness result for classes of actions of many small categories on quantum compact metric spaces by Lipschitz linear maps, for the topology of the covariant Gromov-Hausdorff propinquity. In particular, our result applies to…

We consider a finitely generated virtually abelian group $G$ acting properly and without inversions on a CAT(0) cube complex $X$. We prove that $G$ stabilizes a finite dimensional CAT(0) subcomplex $Y \subseteq X$ that is isometrically…

For all systolic groups we construct boundaries which are EZ--structures. This implies the Novikov conjecture for torsion--free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex,…

Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated…

We survey the known group properties that a sequence of finite groups or group actions needs to satisfy to admit subsets of bounded cardinality producing expander Cayley or Schreier graphs. We prove that an infinite amenable group and…

Let G be the automorphism group of a regular right-angled building X. The "standard uniform lattice" \Gamma_0 in G is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the…

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic…

We provide a necessary and sufficient condition on a finite flag simplicial complex, L, for which there exists a unique CAT(0) cube complex whose vertex links are all isomorphic to L. We then find new examples of such CAT(0) cube complexes…

We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric…

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups $\Gamma$ endowed with a rich boundary. The starting point is the Strip Criterion of identification of the…

In this paper we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish necessary and sufficient conditions for the group to be…

Natural collectives, despite comprising individuals who may not know their numerosity, can exhibit behaviors that depend sensitively on it. This paper proves that the collective behavior of number-oblivious individuals can even have a…

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of…

Gromov's theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. A key ingredient in its proof is the small doubling property. In this work, we study entropy analogues of this property…

A continuous action of a group G on a compact metric space has sensitive dependence on initial conditions if there is a number e>0 such that for any open set U we can find g in G such that g.U has diameter greater than e. We prove that if a…

We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property…

Let $G$ be a discrete group, $\mu$ a measure on $G$ and $X$ a proper CAT(0) space. We show that if $G$ acts non-elementarily with a rank one element on $X$, then the pushforward $\{Z_n o \}_n$ to $X$ of the random walk generated by $\mu$…

A number of properties of spherical Artin groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are…

We prove a Banach version of \.Zuk's criterion for groups acting on partite simplicial complexes. Using this new criterion we derive a new fixed point theorem for random groups in the Gromov density model with respect to several classes of…