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Anthony Genevois

Publications and source records attributed to Anthony Genevois.

At least 19 recordsLinked to original sources
arXiv2026

A median degree from crossing graphs of median graphs

The crossing graph $\mathrm{Cross}(M)$ of a median graph $M$ is defined as the graph whose vertices are the $Θ$-classes of $M$ and whose edges connect two $Θ$-classes whenever they cross. It is known that every graph $X$ can be realised as the crossing graph of some median graph. In this article, we initiate the study of the space $\mathrm{Cross}^{-1}(X)$ of all the median graphs with crossing graph $X$. First, we prove that two finite median graphs have isomorphic crossing graphs if and only if one can be obtained from the other by a sequence of elementary transformations we call slidings. Then, motivated by the fact that $\mathrm{Cross}^{-1}(X)$ always contains a single median graph of maximal degree, namely the simplex-graph of $X$, we introduce the median degree of $X$ as the smallest possible degree of a median graph in $\mathrm{Cross}^{-1}(X)$. We compute the median degree for some families of graphs and characterise the graphs with maximal median degree.

math.CO↗
arXiv2026

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

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 pointed sums $\bigvee_\mathbb{N} Γ_1^{\bowtie}$ and $\bigvee_\mathbb{N} Γ_2^{\bowtie}$ are homotopy equivalent, where $Γ_i^{\bowtie}$ denotes the simplicial complex whose vertex-set is $Γ_i$ and whose simplices are given by joins. These invariants are extracted from a study, of independent interest, of the homotopy types of several complexes of hyperplanes in quasi-median graphs (such as one-skeleta of CAT(0) cube complexes). For instance, given a quasi-median graph $X$, the \emph{crossing complex} $\mathrm{Cross}^\triangle(X)$ is the simplicial complex whose vertices are the hyperplanes (or $θ$-classes) of $X$ and whose simplices are collections of pairwise transverse hyperplanes. When $X$ has no cut-vertex, we show that $\mathrm{Cross}^\triangle(X)$ is homotopy equivalent to the pointed sum of the links of all the vertices in the prism-completion $X^\square$ of $X$.

math.GR↗
arXiv2026

Cell structure of mediangle graphs

Mediangle graphs are a common generalization of median graphs (1-sekeleta of CAT(0) cube complexes) and Coxeter graphs (Cayley graphs of Coxeter systems). Answering a question motivated from geometric group theory, we show that these graphs can be endowed with the structure of a contractible cell complex. We further show that the cells of this complex are products of simplices and simplicial oriented matroids. A crucial part of the proof identifies bipartite mediangle graphs as tope graphs of finitary Complexes of Oriented Matroids.

math.CO↗
arXiv2026

Polynomial hyperbolicity and products of free groups

In this article, we define a locally finite graph $X$ as $η$-polynomially hyperbolic if there exists a Lipschitz map $φ: X \to Z$ to some hyperbolic space $Z$ satisfying the following condition: there exists $C \geq 0$ such that $$|B(p,R_1) \cap φ^{-1} (B(q,R_2))| \leq (C R_1)^{η(C R_2)} \text{ for all } p,q \in X, R_1,R_2 \geq 0.$$ The picture to keep in mind is that coarse fibres of $φ$ have polynomial growth with a degree coarsely controlled by $η$ as the thickness of the fibres grows. The map $η$ quantifies how brutal we have to be in order to turn $X$ into a hyperbolic space. Our main result is that, among cocompact special groups, being $\mathrm{lin}$-polynomially hyperbolic amounts not to contain $\mathbb{F}_2 \times \mathbb{F}_2$ as a subgroup. Consequently, containing $\mathbb{F}_2 \times \mathbb{F}_2$ as a subgroup turns out to be quasi-isometric invariant for cocompact special groups.

math.GR↗
arXiv2026

Relative numbers of ends and quasi-median graphs

Given a finitely generated $G$ and a subgraph $H \leq G$, the relative number of ends $e(G,H)$ is the number of ends of a Schreier graph $\mathrm{Sch}(G,H)$ and the number of coends $\tilde{e}(G,H)$ is the maximal number of $H$-infinite components of the complement of a neighbourhood of $H$ in $G$. Generalising Sageev's characterisation of codimension-one subgroups in terms of actions on CAT(0) cube complexes, we characterise the number of relative ends and the number of coends of a pair $(G,H)$ in terms of actions on quasi-median graphs.

math.GR↗
arXiv2026

Virtual splittings of right-angled Artin groups

In this article, we determine, given a finite graph $Γ$ and an integer $n \geq 1$, when a right-angled Artin group $A(Γ)$ virtually splits over an abelian subgroup of rank $n$. More precisely, we show that the following assertions are equivalent: (1) $A(Γ)$ admits $\mathbb{Z}^n$ as a codimension-one subgroup, (2) $A(Γ)$ virtually splits over $\mathbb{Z}^n$, (3) $A(Γ)$ splits over $\mathbb{Z}^n$, and (4) $Γ$ either is a complete graph with $n+1$ vertices or contains a complete subgraph of size $n$ that has a subgraph separating $Γ$.

math.GR↗
arXiv2026

Coarse separation and splittings in right-angled Artin groups

In this article, we characterise geometrically when a right-angled Artin group splits over an abelian subgroup. More precisely, given a finite graph $Γ$, we show that $A(Γ)$ splits over an abelian subgroup if and only if it is coarsely separable by a family of subexponential growth, which amounts to saying that $Γ$ is complete or separated by a complete subgraph.

math.GR↗
arXiv2026

Coarse separation and splittings in hyperbolic groups

We study coarse separation in one-ended hyperbolic groups from a quantitative point of view, focusing on the volume growth of separating subsets. We prove that a one-ended hyperbolic group that is not virtually a surface group is coarsely separable by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. To do so, we show that sufficiently large thickened spheres are hard to cut, in the sense that their cut-sets have exponential size, a result of independent interest. As an application, we obtain a polynomial lower bound on the separation profile of one-ended hyperbolic groups that do not split over a two-ended subgroup. We also apply our criterion to graph products of finite groups, giving a combinatorial characterisation of when such graph products are coarsely separable by a subset of subexponential growth.

math.GR↗
arXiv2026

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

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 propose a combinatorial geometry (mediangle graphs) to study them. As an application, we give natural and unified proofs for several results that hold for both Coxeter groups and graph products of groups.

math.GR↗
arXiv2025

Examples of cubulable groups with fixed-point properties

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 satisfying $(\mathrm{FW}_n)$ but with good cubical properties in higher dimensions. First, we prove that, for a finitely generated group $G$ with no non-abelian free subgroup, $G$ satisfies $(\mathrm{FW}_n)$ if and only if no subgroup $H \leq G$ of index $\leq n$ can be mapped to $\mathbb{D}_\infty$ with an infinite image. For instance, the affine Coxeter group $\tilde{A}_n$ satisfies $(\mathrm{FW}_n)$ but not $(\mathrm{FW}_{n+1})$. In another direction, we investigate virtually graph products of finite groups. As an application of our constructions, we find explicit examples, for every $n \geq 1$, of acylindrically hyperbolic groups that are cocompactly cubulable but satisfy $(\mathrm{FW}_n)$. Several conjectures and open questions are included.

math.GR↗
arXiv2025

Asymptotically rigid mapping class groups III: Presentations and isomorphisms

This article is dedicated to the computation of an explicit presentation of some asymptotically rigid mapping class groups, namely the braided Higman-Thompson groups. To do so, we use the action of these groups on the spine complex, a simply connected cube complex constructed by the authors in a previous work. In particular, this allows to compute the abelianisations of these groups. With these new algebraic invariants we can handle many new cases of the isomorphism problem for asymptotically rigid mapping class groups of trees.

math.GR↗
arXiv2025

An illustrated introduction to the coarse topology of lamplighters

Roughly speaking, lamplighter graphs encode the possible configurations of a lamplighter that moves along a given graph and that modifies the colours of lamps at vertices. This article is dedicated to the following delicate question: when do two lamplighter graphs have the same coarse geometry, i.e.\ when are they quasi-isometric? Inspired by elementary ideas from topology, which we will ``coarsify'', I will survey some techniques that allow us to compare efficiently lamplighter graphs (and more) up to quasi-isometry. Based on a minicourse given during the Journées de Topologie Géométrique at the Institut Fourier in August 2025.

math.GR↗
arXiv2025

Beyond graph products and cactus groups: quandle products of groups

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 recently introduced trickle groups. Our approach is geometric: we show that quandle products admit quasi-median Cayley graphs; and, then, we exploit this geometry to deduce various valuable information about quandle products.

math.GR↗
arXiv2025

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

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 Coxeter groups, graph products, and Dyer groups. As a consequence, we deduce that, unless they virtually split as direct products, periagroups are acylindrically hyperbolic and their conjugacy growth series, with respect to standard generating sets, are transcendental.

math.GR↗
arXiv2025

Cyclic hyperbolicity in CAT(0) cube complexes

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. Pursuing in this direction, we show that $G$ does not contain $\mathbb{F}_2 \times \mathbb{F}_2$ if and only if it is weakly hyperbolic relative to cyclic subgroups, or cyclically hyperbolic for short. This observation motivates the study of cyclically hyperbolic groups, which we initiate in the class of groups acting geometrically on CAT(0) cube complexes. Given such a group $G$, we first prove a structure theorem: $G$ virtually splits as the direct sum of a free abelian group and an acylindrically hyperbolic cubulable group. Next, we prove a strong Tits alternative: every subgroup $H \leq G$ either is virtually abelian or it admits a series $H=H_0 \rhd H_1 \rhd \cdots \rhd H_k$ where $H_k$ is acylindrically hyperbolic and where $H_i/H_{i+1}$ is finite or free abelian. As a consequence, $G$ is SQ-universal and it cannot contain subgroups such that products of free groups and virtually simple groups.

math.GR↗
arXiv2025

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

For every $n\geq 1$, the flat braid group $\mathrm{FB}_n$ is an analogue of the braid group $B_n$ that can be described as the fundamental group of the configuration space $$\left\{ \{x_1, \ldots, x_n \} \in \mathbb{R}^n / \mathrm{Sym}(n) \mid \text{there exist at most two indices $i,j$ such that } x_i=x_j \right\}.$$ Alternatively, $\mathrm{FB}_n$ can also be described as the right-angled Coxeter group $C(P_{n-2}^\mathrm{opp})$, where $P_{n-2}^\mathrm{opp}$ denotes the opposite graph of the path $P_{n-2}$ of length $n-2$. In this article, we prove that, for every $n= 7$ or $\geq 11$, $\mathrm{PFB}_n$ is not virtually a right-angled Artin group, disproving a conjecture of Naik, Nanda, and Singh. In the opposite direction, we observe that $\mathrm{FB}_7$ turns out to be commensurable to the right-angled Artin group $A(P_4)$.

math.GR↗
arXiv2025

Hyperbolic structures on Houghton groups

Given a group $G$, its poset of hyperbolic structures $\mathcal{H}(G)$ encodes all the possible cobounded actions of $G$ on hyperbolic spaces. In this article, we describe the poset $\mathcal{H}(H_n)$ for every Houghton group $H_n$, $n \geq 2$. In particular, we show that $H_n$ admits exactly $n$ focal hyperbolic structures. As an application, we construct the first example of a group admitting exactly one focal hyperbolic structure, answering a question of Abbott, Balasubramanya, and Osin.

math.GR↗