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Anthony Genevois

Publications and source records attributed to Anthony Genevois.

67 records · Page 4Linked to original sources
arXiv2019

Hyperbolicities in CAT(0) cube complexes

This paper is a survey dedicated to the following question: given a group acting on some CAT(0) cube complex, how to exploit this action to determine whether or not the group is Gromov / relatively / acylindrically hyperbolic? As much as possible, the different criteria we mention are illustrated by applications. We also propose a model for universal acylindrical actions of cubulable groups and give a few applications to Morse, stable and hyperbolically embedded subgroups.

math.GR↗
arXiv2019

On the geometry of van Kampen diagrams of graph products of groups

In this article, we propose a geometric framework dedicated to the study of van Kampen diagrams of graph products of groups. As an application, we find information on the word and the conjugacy problems. The main new result of the paper deals with the computation of conjugacy length functions. More precisely, if $Γ$ is a finite graph and $\mathcal{G}= \{ G_u \mid u \in V(Γ) \}$ a collection of finitely generated groups indexed by the vertices of $Γ$, then $$\max\limits_{u \in V(Γ)} \mathrm{CLF}_{G_u}(n) \leq \mathrm{CLF}_{Γ\mathcal{G}}(n) \leq (D+1) \cdot n + \max\limits_{Δ\subset Γ\ \text{complete}} \sum\limits_{u \in V(Δ)} \mathrm{CLF}_{G_u}(n)$$ for every $n \geq 1$, where $D$ denotes the maximal diameter of a connected component of the opposite graph $Γ^{\mathrm{opp}}$. As a consequence, a graph product of groups with linear conjugacy length functions has linear conjugacy length function as well.

math.GR↗
arXiv2018

Partially CAT(-1) groups are acylindrically hyperbolic

In this paper, we show that, if a group $G$ acts geometrically on a geodesically complete CAT(0) space $X$ which contains at least one point with a CAT(-1) neighborhood, then $G$ must be either virtually cyclic or acylindrically hyperbolic. As a consequence, the fundamental group of a compact Riemannian manifold whose sectional curvature is nonpositive everywhere and negative in at least one point is either virtually cyclic or acylindrically hyperbolic. This statement provides a precise interpretation of an idea expressed by Gromov in his paper Asymptotic invariants of infinite groups.

math.GR↗
arXiv2018

A note on nilpotent subgroups of automorphism groups of RAAGs

We observe that automorphism groups of right-angled Artin groups contain nilpotent non-abelian subgroups, namely $H_3(\mathbb{Z})$ the three-dimensional integer Heisenberg group, provided they admit a certain type of element, called an adjacent transvection. This represents a (minor) extension of a result of Charney-Vogtmann.

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arXiv2018

Automorphism groups of cyclic products of groups

This article initiates a geometric study of the automorphism groups of general graph products of groups, and investigates the algebraic and geometric structure of automorphism groups of cyclic product of groups. For a cyclic product of at least five groups, we show that the action of the cyclic product on its Davis complex extends to an action of the whole automorphism group. This action allows us to completely compute the automorphism group and to derive several of its properties: Tits Alternative, acylindrical hyperbolicity, lack of property (T).

math.GR↗
arXiv2017

Cubical-like geometry of quasi-median graphs and applications to geometric group theory

The class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the definition of hyperplanes from CAT(0) cube complexes, and we show that the geometry of a quasi-median graph essentially reduces to the combinatorics of its hyperplanes. In the second part, we exploit the specific structure of the hyperplanes to state combination results. The main idea is that if a group acts in a suitable way on a quasi-median graph so that clique-stabilisers satisfy some non-positively curved property $\mathcal{P}$, then the whole group must satisfy $\mathcal{P}$ as well. The properties we are interested in are mainly (relative) hyperbolicity, (equivariant) $\ell^p$-compressions, CAT(0)-ness and cubicality. In the third part, we apply our general criteria to several classes of groups, including graph products, Guba and Sapir's diagram products, some wreath products, and some graphs of groups. Graph products are our most natural examples, where the link between the group and its quasi-median graph is particularly strong and explicit; in particular, we are able to determine precisely when a graph product is relatively hyperbolic.

math.GR↗
arXiv2017

Groups acting on quasi-median graphs. An introduction

Quasi-median graphs have been introduced by Mulder in 1980 as a generalisation of median graphs, known in geometric group theory to naturally coincide with the class of CAT(0) cube complexes. In his PhD thesis, the author showed that quasi-median graphs may be useful to study groups as well. In the present paper, we propose a gentle introduction to the theory of groups acting on quasi-median graphs.

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arXiv2016

Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups

The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and we show that contracting isometries are strongly related to isolated points at infinity, when the complex is locally finite. This boundary turns out to appear naturally in the context of Guba and Sapir's diagram groups, and we apply our main criterion to determine precisely when an element of a diagram group induces a contracting isometry on the associated Farley cube complex. As a consequence, in some specific case, we are able to deduce a criterion to determine precisely when a diagram group is acylindrically hyperbolic.

math.GR↗
arXiv2016

Coning-off CAT(0) cube complexes

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups, which is a particular case of a result due to Behrstock, Caprace and Hagen. A second application gives the acylindrical hyperbolicity of $C'(1/4)-T(4)$ small cancellation quotients of free products.

math.GR↗
arXiv2015

Hyperplanes of Squier's cube complexes

To any semigroup presentation $\mathcal{P}= \langle Σ\mid \mathcal{R} \rangle$ and base word $w \in Σ^+$ may be associated a nonpositively curved cube complex $S(\mathcal{P},w)$, called a Squier complex, whose underlying graph consists of the words of $Σ^+$ equal to $w$ modulo $\mathcal{P}$ where two such words are linked by an edge when one can be transformed into the other by applying a relation of $\mathcal{R}$. A group is a diagram group if it is the fundamental group of a Squier complex. In this paper, we describe hyperplanes in these cube complexes. As a first application, we determine exactly when $S(\mathcal{P},w)$ is a special cube complex, as defined by Haglund and Wise, so that the associated diagram group embeds into a right-angled Artin group. A particular feature of Squier complexes is that the intersections of hyperplanes are "ordered" by a relation $\prec$. As a strong consequence on the geometry of $S(\mathcal{P},w)$, we deduce, in finite dimensions, that its univeral cover isometrically embedds into a product of finitely-many trees with respect to the combinatorial metrics; in particular, we notice that (often) this allows to embed quasi-isometrically the associated diagram group into a product of finitely-many trees. Finally, we exhibit a class of hyperplanes inducing a decomposition of $S(\mathcal{P},w)$ as a graph of spaces, and a fortiori a decomposition of the associated diagram group as a graph of groups, giving a new method to compute presentations of diagram groups. As an application, we associate a semigroup presentation $\mathcal{P}(Γ)$ to any finite interval graph $Γ$, and we prove that the diagram group associated to $\mathcal{P}(Γ)$ (for a given base word) is isomorphic to the right-angled Artin group $A(\overlineΓ)$.

math.GR↗
arXiv2015

Hyperbolic diagram groups are free

In this paper, we study the so-called diagram groups. Our main result is that diagram groups are free if and only if they do not contain any subgroup isomorphic to $\mathbb{Z}^2$. As an immediate corollary, we get that hyperbolic diagram groups are necessarily free, answering a question of Guba and Sapir.

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