Related papers: CAT(0) and cubulated Shephard groups

The $\widetilde{A}_n$ Coxeter groups are known to not be systolic or cocompactly cubulated for $n\geq 3$. We prove that these groups act geometrically on weakly modular graphs, a weak notion of nonpositive curvature generalizing the…

For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct explicit examples of groups…

In this note, we discuss and motivate the use of the terminology ``median graphs'' in place of ``CAT(0) cube complexes'' in geometric group theory.

The aim of this thesis is to present the notion of spaces whose curvature is bounded above, and to give some of its application in the context of Combinatorial Algebra. The thesis is made of two parts, one of theoretic purpose, and the…

The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked…

In this article, we introduce rotation groups as a common generalisation of Coxeter groups and graph products of groups (including right-angled Artin groups). We characterise algebraically these groups by presentations (periagroups) and we…

We show that plane Cremona groups over finite fields embed as dense subgroups into Neretin groups, i.e. groups of almost automorphisms of rooted trees. We also show that if the finite base field has even characteristic and contains at least…

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and…

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 a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that $G$ is a group for which there is a bound on the orders of its finite subgroups. We prove that if $G$ acts properly on a…

The notions of nonpositive curved spaces and biautomatic groups are generalizations of the geometric properties of hyperbolic spaces and computational properties of their fundamental groups. Given the mutual origins of these conditions, one…

We study uniform exponential growth of groups acting on CAT(0) cube complexes. We show that groups acting without global fixed points on CAT(0) square complexes either have uniform exponential growth or stabilize a Euclidean subcomplex.…

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…

We give a uniform construction of the finite simple groups $E_6(q)$, $F_4(q)$ and ${}^2E_6(q)$, which does not require any special treatment for characteristics 2 or 3, and in particular avoids any mention of quadratic Jordan algebras.…

We prove that if a group $G$ admits a virtually special action on a CAT(0) cube complex, then any product of convex-cocompact subgroups of $G$ is separable. Previously, this was only known for products of three subgroups, or in the case…

It is known that a cocompact special group $G$ does not contain $\mathbb{Z} \times \mathbb{Z}$ if and only if it is hyperbolic; and it does not contain $\mathbb{F}_2 \times \mathbb{Z}$ if and only if it is toric relatively hyperbolic.…

For a finite group action on a finite EI quiver, we construct its `orbifold' quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action.…

In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that…

In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as \emph{median sets}, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are…

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension $d$ for all $d\geq 4$. These examples result from applying CAT$(0)$/CAT$(-1)$ filling constructions (based on singular doubly warped products) to…