Related papers: Uniform exponential growth for CAT(0) square compl…
We show that the Hilbert space compression of any finite dimensional CAT(0) cube complex is 1 and deduce that any discrete group acting properly, co-compactly on a CAT(0) cube complex is exact. The class of groups covered by this theorem…
We prove that if $G = G_1\times\dots\times G_n$ acts essentially, properly and cocompactly on a CAT(0) cube complex X, then the cube complex splits as a product. We use this theorem to give various examples of groups for which the minimal…
We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0)-space. This…
One way to show that Thompson's group F is non-amenable is to exhibit an action of F on a locally compact CAT(0) space X containing no F-invariant flats and having no global fixed points in its boundary-at-infinity. We study the actions of…
We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube…
We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats and the Relative Fellow Traveller Property. The groups in question include word hyperbolic CAT(0) groups as well as…
We prove that a random group has fixed points when it isometrically acts on a CAT(0) cube complex. We do not assume that the action is simplicial.
In [4], Dunwoody defined resolutions for finitely presented group actions on simplicial trees, that is, an action of the group on a tree with smaller edge and vertex stabilizers. He, moreover, proved that the size of the resolution is…
In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with…
An expansion set is a set $\mathcal{B}$ such that each $b \in \mathcal{B}$ is equipped with a set of expansions $\mathcal{E}(b)$. The theory of expansion sets offers a systematic approach to the construction of classifying spaces for…
Our main result is that for densities $<\frac{3}{10}$ a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality, to some class of non-planar diagrams and,…
The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and…
We present a new and very short proof of the fact that, for positive $C_0$-semigroups on spaces of continuous functions, the spectral and the growth bound coincide. Our argument, inspired by an idea of Vogt, makes the role of the underlying…
Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize 2-dimensional CAT(0) cube complexes and are a square analog of systolic complexes. We introduce and study…
We generalize ideas of Jahncke from trees to square complexes. We introduce the notion of progression in $\mathrm{CAT}(0)$ square complexes. Using progression, we are able to build on the proof strategy of Dahmani-Guirardel-Przytycki to…
For n greater or equal 4 we discuss questions concerning global fixed points for isometric actions of Aut(Fn), the automorphism group of a free group of rank n, on complete CAT(0) spaces. We prove that whenever Aut(Fn) acts by isometries on…
In this paper we obtain uniform positive lower bounds on stable commutator length in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated…
In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at…
We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite…
The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the…