A. Genevois

Beyond graph products and cactus groups: quandle products of groups

Anthony Genevois

In this paper, we introduce and initiate the study of quandle products of groups, a family of groups that includes graph products of groups, cactus groups, wreath products, and the…

Contracting elements and conjugacy growth in Coxeter groups, graph products, and further groups

Laura Ciobanu, Anthony Genevois

In this article we construct contracting elements in the standard Cayley graphs of the so-called periagroups, a family of groups introduced by the second-named author which include…

Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups

Carolyn Abbott, Anthony Genevois, Eduardo Martinez-Pedroza

In this article, we prove that, given two finite connected graphs $Γ_{1}$ and $Γ_{2}$ , if the two right-angled Artin groups $A (Γ_{1})$ and $A (Γ_{2})$ are quasi-isometric, then the infinite…

Flat braid groups, right-angled Artin groups, and commensurability

Anthony Genevois

For every $n \geq 1$ , the flat braid group $FB_{n}$ is an analogue of the braid group $B_{n}$ that can be described as the fundamental group of the configuration space $$\left\…

Hyperbolic structures on Houghton groups

Anthony Genevois, Geoffrey Tournier

Given a group $G$ , its poset of hyperbolic structures $H (G)$ encodes all the possible cobounded actions of $G$ on hyperbolic spaces. In this article, we describe the pose…

math.GR2024

Rotation groups virtually embed into right-angled rotation groups

Anthony Genevois

It is a theorem due to F. Haglund and D. Wise that reflection groups (aka Coxeter groups) virtually embed into right-angled reflection groups (aka right-angled Coxeter groups). In…

math.GR2023

Examples of cubulable groups with fixed-point properties

Anthony Genevois

For every $n \geq 1$ , let $(FW_{n})$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$ )…

math.GR2023

Right-angled Artin groups are symmetric diagram groups

Paolo Bellingeri, Anthony Genevois, Neha Nanda

In this article, we show that, for every $n \geq 2$ , the pure virtual twin group $P V T_{n}$ can be naturally described as a symmetric diagram group, a family of groups introduced by V…

math.GR2023

Embedding problem between right-angled Artin groups

Anthony Genevois

Given a graph $Γ$ , the right-angled Artin group $A (Γ)$ is given by the presentation $⟨ u \in V (Γ) ∣ [u, v] = 1, {u, v} \in E (Γ)⟩$ . The Embedding Problem in right-an…

Cactus groups from the viewpoint of geometric group theory

Anthony Genevois

Cactus groups and their pure subgroups appear in various fields of mathematics and are currently attracting attention from diverse mathematical communities. They share similarities…

Property (NL) for group actions on hyperbolic spaces

Sahana Balasubramanya, Francesco Fournier-Facio, Anthony Genevois +1

We introduce Property (NL), which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. In other words, such a group $G$ can…

Rotation groups, mediangle graphs, and periagroups: a unified point of view on Coxeter groups and graph products of groups

Anthony Genevois

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 algeb…

An introduction to diagram groups

Anthony Genevois

To every semigroup presentation $P = ⟨ Σ ∣ R ⟩$ and every baseword $w \in Σ^{+}$ can be associated a diagram group $D (P, w)$ , defined as the…

Translation lengths in crossing and contact graphs of (quasi-)median graphs

Anthony Genevois

Given a quasi-median graph $X$ , the crossing graph $Δ X$ and the contact graph $Γ X$ are natural hyperbolic models of $X$ . In this article, we show that the asymptotic translation le…

No quasi-isometric rigidity for proper actions on CAT(0) cube complexes

Francesco Fournier-Facio, Anthony Genevois

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 admi…

Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

Anthony Genevois, Anne Lonjou, Christian Urech

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 fin…

A note on morphisms to wreath products

Anthony Genevois, Romain Tessera

Given a morphism $φ : G \to A ≀ B$ from a finitely presented group $G$ to a wreath product $A ≀ B$ , we show that, if the image of $φ$ is a sufficiently large subgroup, then $\ma…

Asymptotically rigid mapping class groups II: strand diagrams and nonpositive curvature

Anthony Genevois, Anne Lonjou, Christian Urech

In this article, we introduce a new family of groups, called Chambord groups and constructed from braided strand diagrams associated to specific semigroup presentations. It include…

Cyclic hyperbolicity in CAT(0) cube complexes

Anthony Genevois

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

math.GR2021★ 3 cited

Asymptotic geometry of lamplighters over one-ended groups

Anthony Genevois, Romain Tessera

This article is dedicated to the asymptotic geometry of wreath products $F ≀ H := (⨁_{H} F) ⋊ H$ where $F$ is a finite group and $H$ a one-ended finitely…