Related papers: Geodesic growth of right-angled Coxeter groups bas…

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…

A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every…

We introduce right generating sets, Cayley graphs, growth functions, types and rates, and isoperimetric constants for left homogeneous spaces equipped with coordinate systems; characterise right amenable finitely right generated left…

The space $\mathcal{G}_n$ of $n$-marked groups provides a natural framework for studying algebraic and geometric invariants under deformation. In general, the growth rate is not continuous on $\mathcal{G}_n$. In this paper, we investigate…

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n <…

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

We study product set growth in groups with acylindrical actions on quasi-trees and, more generally, hyperbolic spaces. As a consequence, we show that for every surface $S$ of finite type, there exist $\alpha,\beta>0$ such that for any…

Let $A$ be an associative algebra graded by a finite group $G$ over a field ${F}$ of characteristic zero. One associates to $A$ the sequence of $G$-graded codimensions $c_n^G(A)$, $n=1,2,\ldots$, which measures the growth of the polynomial…

A set D of vertices of a graph G with vertex set V is irredundant if each non-isolated vertex of G[D] has a neighbour in V-D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an…

We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using deceasing induction on $ \mathbf{a} $-values and a kind of decomposition formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of…

In this paper we consider groups of the form $G\wr L$, where the set of generators naturally extends the sets of generators of $G$ and $L$, and $L$ admits a Cayley graph that is a tree. We show how one can compute the conjugacy growth…

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs…

High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of…

This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph…

Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices,…

We study Morse subgroups and Morse boundaries of random right-angled Coxeter groups in the Erd\H{o}s--R\'enyi model. We show that at densities below $\left(\sqrt{\frac{1}{2}}-\epsilon\right)\sqrt{\frac{\log{n}}{n}}$ random right-angled…

We show that the growth of a unimodular random rooted tree $(T,o)$ of degree bounded by $d$ always exists, assuming its upper growth passes the critical threshold $\sqrt{d-1}$. This complements Timar's work who showed the possible…

We introduce the notion of city products of right-angled buildings that produces a new right-angled building out of smaller ones. More precisely, if $M$ is a right-angled Coxeter diagram of rank $n$ and $\Delta_1,\dots,\Delta_n$ are…

Let G be a semisimple Lie group with associated symmetric space D, and let Gamma subset G be a cocompact arithmetic group. Let L be a lattice inside a Z Gamma-module arising from a rational finite-dimensional complex representation of G.…

Complex systems have motivated continuing interest from the scientific community, leading to new concepts and methods. Growing systems represent a case of particular interest, as their topological, geometrical, and also dynamical properties…