Related papers: Limit groups are CAT(0)

We show that, given any finite dimensional, connected, compact metric space Z, there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry f from X to Y, and a geodesic ray c in X, such that the…

We consider actions of locally compact groups $G$ on certain CAT(0) spaces $X$ by isometries. The CAT(0) spaces we consider have finite dimension at large scale. In case $B$ is a $G$-boundary, that is a measurable $G$-space with amenability…

If G is a group acting properly by semisimple isometries on a proper CAT(0) space X, then we build models for the classifying spaces E_{vc} and E_{fbc} under the additional assumption that the action of G has a well-behaved collection of…

For each $n$ we construct examples of finitely presented $C'(1/6)$ small cancellation groups that do not act properly on any $n$-dimensional CAT(0) cube complex.

We study lattices acting on $\mathrm{CAT}(0)$ spaces via their commensurated subgroups. To do this we introduce the notions of a graph of lattices and a complex of lattices giving graph and complex of group splittings of $\mathrm{CAT}(0)$…

We explain why semistability of a one-ended proper CAT(0) space can be determined by the geodesic rays. This is applied to boundaries of CAT(0) groups.

In this paper, we study boundary actions of CAT(0) spaces from a point of view of topological dynamics and $C^*$-algebras. First, we investigate the actions of right-angled Coexter groups and right-angled Artin groups with finite defining…

We show that group actions on irreducible ${\rm CAT(0)}$ cube complexes with no free faces are uniquely determined by their $\ell^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and…

It is proved that the mapping class group of any closed surface with finitely many marked points is quasiisometric to a CAT(0) cube complex. We provide two distinct proofs, one tailored to mapping class groups, and one applying to a larger…

We prove that for $n\geq 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, in the sense of Einstein and Groves. As a consequence, if $\Gamma$ is a non-uniform…

We prove that all finite graphs of groups with cyclic vertex and edge groups act freely and isometrically on a complete, nonpositively curved geodesic metric space.

This paper is about geometric and topological properties of a proper CAT(0) space $X$ which is cocompact - i.e. which has a compact generating domain with respect to the full isometry group. It is shown that geodesic segments in $X$ can…

We show, under mild hypotheses, that if each element of a finitely generated group acting on a $2$-dimensional $\mathrm{CAT}(0)$ complex has a fixed point, then there is a global fixed point. In particular all actions of finitely generated…

We provide conditions on the defining graph of a right-angled Coxeter group presentation that guarantees the boundary of any CAT(0) space on which the group acts geometrically will be locally connected. This is a revised version of a…

Shephard groups are common generalizations of Coxeter groups, Artin groups, and graph products of cyclic groups. Their definition is similar to that of a Coxeter group, but generators may have arbitrary order rather than strictly order 2.…

A group is tubular if it acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. We prove that a tubular group is virtually special if and only if it acts freely on a locally finite CAT(0) cube complex.…

The main result of this paper is that given a group $G$ acting geometrically by isometries on a CAT(0) space $X$ and a cyclic subgroup $H$ of $G$ generated by a rank-1 isometry of $X$, $H$ has bounded packing in $G$. We give two proofs of…

We exhibit a variety of groups that act properly and even cocompactly on median graphs (a.k.a. one-skeletons of CAT(0) cube complexes), with quasi-isometric groups that do not admit any proper action on a median graph. This answes a…

We give sufficient conditions for a group acting on a geodesic metric space to be acylindrically hyperbolic and mention various applications to groups acting on CAT($0$) spaces. We prove that a group acting on an irreducible non-spherical…

Let $\Gamma$ be a finitely generated group acting properly discontinuously by isometries on a visibility CAT(0) space $X$ that satisfies the bounded packing property. We prove that $\Gamma$ satisfies the Tits alternative: it is either…