Related papers: A generalization of the Davis-Moussong complex for…

Coxeter groups are equipped with a partial order known as the weak order, such that $u \leq v$ if the inversions of $u$ are a subset of the inversions of $v$. In finite Coxeter groups, weak order is a complete lattice, but in infinite…

We prove the dichotomy that every Coxeter group either has a strongly solid group von Neumann algebra or contains the product of an infinite cyclic group and a free group of rank 2. This generalizes the same dichotomy for right-angled…

To every semigroup presentation $\mathcal{P}= \langle \Sigma \mid \mathcal{R} \rangle$ and every baseword $w \in \Sigma^+$ can be associated a diagram group $D(\mathcal{P},w)$, defined as the fundamental group of the so-called Squier…

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of…

We prove the Tits Alternative for groups acting on $2$-dimensional $\mathrm{CAT}(0)$ complexes with a bound on the order of the cell stabilisers.

Burger and Mozes (1997) constructed the first examples of simple uniform lattices in products of trees. In this paper, we construct simple uniform lattices in products of certain Davis complexes. More precisely, we consider lattices in…

In this article we study the asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. We present a family of CAT(0) cube complexes on which the latter groups act. Along the way, we…

Shephard groups are common extensions of Artin and Coxeter groups. They appear, for example, in algebraic study of manifolds. An infinite family of Shephard groups which are not Artin or Coxeter groups is considered. Using techniques form…

We provide a systematic description of the automorphism groups of specially cocompact CAT(0) cube complexes. We show that these groups are topologically finitely generated, present a method to explicitly obtain generating sets, and prove a…

A Coxeter group of classical type $A_n$, $B_n$ or $D_n$ contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what…

Let $(W,S)$ be a Coxeter system with Davis complex $\Sigma$. The polyhedral automorphism group $G$ of $\Sigma$ is a locally compact group under the compact-open topology. If $G$ is a discrete group (as characterised by Haglund--Paulin),…

We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups…

In this paper, we describe a regular representation given by Cayley theorem for 2-crossed modules of groups and their associated Gray 3-group groupoids with a single 0-cell and equivalently cat2-groups.

The descent algebra $\Sigma(W)$ is a subalgebra of the group algebra $\Q W$ of a finite Coxeter group $W$, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of $W$. Thus $\Sigma(W)$ is a basic…

The purpose of this article is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs $\{\Gamma_i\}_{i\in\mathbb{N}}$ forms a family of expander graphs if and…

In 2011 Eriko Hironaka introduced an interesting generalization of Coxeter groups, motivated by studying certain mapping classes. The generalization is by labeling the vertices of a Coxeter graph either by +1 or by -1, and then generalizing…

We show that for $n \geq 3$, there are torsion-free CAT(0) groups with visual boundaries that embed into $S^n$ but which are not virtually the fundamental group of a compact aspherical $n+1$-manifold. The groups are the CAT(0) and not…

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 build actions of Thompson group V (related to the Cantor set) and of the so-called "spheromorphism" group of Neretin, on "towers" of moduli spaces of genus zero real stable curves. The latter consist of inductive limits of spaces which…

Motivated by the enhanced gauge symmetry phenomenon of the physics literature and mirror symmetry, this paper constructs an action of an Artin group on the derived category of coherent sheaves of a smooth quasiprojective threefold…