arXiv subjects

Anthony Genevois

Publications and source records attributed to Anthony Genevois.

At least 55 records · Page 3Linked to original sources

arXiv2021

Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups

We prove that the automorphism group of every infinitely-ended finitely generated group is acylindrically hyperbolic. In particular $\mathrm{Aut}(\mathbb{F}_n)$ is acylindrically hyperbolic for every $n\ge 2$. More generally, if $G$ is a group which is not virtually cyclic, and hyperbolic relative to a finite collection $\mathcal{P}$ of finitely generated proper subgroups, then $\mathrm{Aut}(G,\mathcal{P})$ is acylindrically hyperbolic. As a consequence, a free-by-cyclic group $\mathbb{F}_n\rtimes_φ\mathbb{Z}$ is acylindrically hyperbolic if and only if $φ$ has infinite order in $\mathrm{Out}(\mathbb{F}_n)$.

math.GR↗

arXiv2021

Asymptotic geometry of lamplighters over one-ended groups

This article is dedicated to the asymptotic geometry of wreath products $F\wr H := \left( \bigoplus_H F \right) \rtimes H$ where $F$ is a finite group and $H$ a one-ended finitely presented group. Our main result is a complete classification of these groups up to quasi-isometry. More precisely, given two finite groups $F_1,F_2$ and two finitely presented one-ended groups $H_1,H_2$, we show that $F_1 \wr H_1$ and $F_2 \wr H_2$ are quasi-isometric if and only if either (i) $H_1,H_2$ are non-amenable quasi-isometric groups and $|F_1|,|F_2|$ have the same prime divisors, or (ii) $H_1,H_2$ are amenable, $|F_1|=k^{n_1}$ and $|F_2|=k^{n_2}$ for some $k,n_1,n_2 \geq 1$, and there exists a quasi-$(n_2/n_1)$-to-one quasi-isometry $H_1 \to H_2$. The article also contains algebraic information on groups quasi-isometric to such wreath products. This can be seen as far reaching extension of a celebrated work of Eskin-Fisher-Whyte who treated the case of $H=\mathbb{Z}$. Our approach is however fundamentally different, as it crucially exploits the assumption that $H$ is one-ended. Our central tool is a new geometric interpretation of lamplighter groups involving natural families of quasi-median spaces.

math.GR↗

arXiv2021

Measure-scaling quasi-isometries

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$κ$-to-one in a natural sense for some $κ>0$. For non-amenable graphs, all quasi-isometries are quasi-$κ$-to-one for any $κ>0$, while for amenable ones there exists at most one possible such $κ$. For an amenable graph $X$, we show that the set of possible $κ$ forms a subgroup of $\mathbb{R}_{>0}$ that we call the (measure-)scaling group of $X$. This group is invariant under measure-scaling quasi-isometries. In the context of Cayley graphs, this implies for instance that two uniform lattices in a given locally compact group have same scaling groups. We compute the scaling group in a number of cases. For instance it is all of $\mathbb{R}_{>0}$ for lattices in Carnot groups, SOL or solvable Baumslag Solitar groups, but is a (strict) subgroup $\mathbb{Q}_{>0}$ for lamplighter groups over finitely presented amenable groups.

math.GR↗

arXiv2021

Lamplighter groups, median spaces, and Hilbertian geometry

From any two median spaces $X,Y$, we construct a new median space $X \circledast Y$, referred to as the diadem product of $X$ and $Y$, and we show that this construction is compatible with wreath products in the following sense: given two finitely generated groups $G,H$ and two (equivariant) coarse embeddings into median spaces $X,Y$, there exist a(n equivariant) coarse embedding $G\wr H \to X \circledast Y$. As an application, we prove that $$α_1(G \wr H) \geq \min(α_1(G),α_1(H))/2 \text{ for all finitely generated groups $G,H$,}$$ where $α_1(\cdot)$ denotes the $\ell^1$-compression. As an other consequence, we recover several well-known theorems related to the Hilbertian geometry of wreath products from a unified point of view: the characterisation of wreath products satisfying Kazhdan's property (T) or the Haagerup property, as well as their discrete versions (FW) and (PW).

math.GR↗

arXiv2020

Median sets of isometries in CAT(0) cube complexes and some applications

In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as \emph{median sets}, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are deduced, including a cubulation of centralisers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a CAT(0) cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on CAT(0) cube complexes.

math.GR↗

arXiv2020

Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson's and Houghton's groups

This article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^\sharp,T^\ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $\mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjian's and Degenhardt's conjectures by showing that $T^\sharp,T^\ast$ are of type $F_\infty$ and that $\mathrm{br}H_n$ is of type $F_{n-1}$ but not of type $F_n$.

math.GR↗

arXiv2020

CAT(0) cube complexes with flat hyperplanes

In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.

math.GR↗

arXiv2020

Special cube complexes revisited: a quasi-median generalisation

In this article, we generalise Haglund and Wise's theory of special cube complexes to groups acting on quasi-median graphs. More precisely, we define special actions on quasi-median graphs, and we show that a group which acts specially on a quasi-median graph with finitely many orbits of vertices must embed as a virtual retract into a graph product of finite extensions of clique-stabilisers. In the second part of the article, we apply the theory to fundamental groups of some graphs of groups called right-angled graphs of groups.

math.GR↗

arXiv2019

Algebraic characterisation of relatively hyperbolic special groups

This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise's special groups. More precise, we introduce a new combinatorial formalism to study (virtually) special groups, and we prove that, given a cocompact special group $G$ and a finite collection of subgroups $\mathcal{H}$, then $G$ is hyperbolic relative to $\mathcal{H}$ if and only if (i) each subgroup of $\mathcal{H}$ is convex-cocompact, (ii) $\mathcal{H}$ is an almost malnormal collection, and (iii) every non-virtually cyclic abelian subgroup of $G$ is contained in a conjugate of some group of $\mathcal{H}$. As an application, we show that a virtually cocompact special group is hyperbolic relative to abelian subgroups if and only if it does not contain $\mathbb{F}_2 \times \mathbb{Z}$.

math.GR↗

arXiv2019

Negative curvature in graph braid groups

In this article, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous article, we determine precisely when a graph braid group is Gromov-hyperbolic, toral relatively hyperbolic, and acylindrically hyperbolic.

math.GR↗

arXiv2019

Morphisms between right-angled Coxeter groups and the embedding problem in dimension two

In this article, given two finite simplicial graphs $Γ_1$ and $Γ_2$, we state and prove a complete description of the possible morphisms $C(Γ_1) \to C(Γ_2)$ between the right-angled Coxeter groups $C(Γ_1)$ and $C(Γ_2)$. As an application, assuming that $Γ_2$ is triangle-free, we show that, if $C(Γ_1)$ is isomorphic to a subgroup of $C(Γ_2)$, then the ball of radius $8|Γ_1||Γ_2|$ in $C(Γ_2)$ contains the basis of a subgroup isomorphic to $C(Γ_1)$. This provides an algorithm determining whether or not, among two given two-dimensional right-angled Coxeter groups, one is isomorphic to a subgroup of the other.

math.GR↗

arXiv2019

Quasi-isometrically rigid subgroups in right-angled Coxeter groups

In the spirit of peripheral subgroups in relatively hyperbolic groups, we exhibit a simple class of quasi-isometrically rigid subgroups in graph products of finite groups, which we call eccentric subgroups. As an application, we prove that, if two right-angled Coxeter groups $C(Γ_1)$ and $C(Γ_2)$ are quasi-isometric, then for any minsquare subgraph $Λ_1 \leq Γ_1$ there exists a minsquare subgraph $Λ_2 \leq Γ_2$ such that the right-angled Coxeter groups $C(Λ_1)$ and $C(Λ_2)$ are quasi-isometric as well. Various examples of non-quasi-isometric groups are deduced. Our arguments are based on a study of non-hyperbolic Morse subgroups in graph products of finite groups. As a by-product, we are able to determine precisely when a right-angled Coxeter group has all its infinite-index Morse subgroups hyperbolic, answering a question of Russell, Spriano and Tran.

math.GR↗

arXiv2019

Negative curvature in automorphism groups of one-ended hyperbolic groups

In this article, we show that some negative curvature may survive when taking the automorphism group of a finitely generated group. More precisely, we prove that the automorphism group $\mathrm{Aut}(G)$ of a one-ended hyperbolic group $G$ turns out to be acylindrically hyperbolic. As a consequence, given a group $H$ and a morphism $φ: H \to \mathrm{Aut}(G)$, we deduce that the semidirect product $G \rtimes_φH$ is acylindrically hyperbolic if and only if $\mathrm{ker}(H \oversetφ{\to} \mathrm{Aut}(G) \to \mathrm{Out}(G))$ is finite.

math.GR↗

arXiv2019

Acylindrical hyperbolicity from actions on CAT(0) cube complexes: a few criteria

The question which motivates the article is the following: given a group acting on a CAT(0) cube complex, how can we prove that it is acylindrically hyperbolic? Keeping this goal in mind, we show a weak acylindricity of the action on the contact graph associated to a CAT(0) cube complex, and we prove a characterisation of WPD contracting isometries of CAT(0) cube complexes. As a first consequence, we find alternative arguments to show several criteria which were proved by Indira Chatterji and Alexandre Martin. Next, we show that if a group acts essentially and acylindrically on the hyperplanes (i.e., the intersection of the stabilisers of two hyperplanes which are sufficiently far away from each other has its cardinality uniformly bounded) on a finite-dimensional CAT(0) cube complex, then it contains a finite-index subgroup which is either acylindrically hyperbolic or cyclic. Finally, we prove a statement about codimension-one subgroups, which implies that, if a group contains a finitely generated codimension-one subgroup which is malnormal and which satisfies the bounded packing property, then this group must contain a finite-index subgroup which is either acylindrically hyperbolic or cyclic.

math.GR↗

arXiv2019

Embeddings into Thompson's groups from quasi-median geometry

The main result of this article is that any braided (resp. annular, planar) diagram group $D$ splits as a short exact sequence $1 \to R \to D \to S \to 1$ where $R$ is a subgroup of some right-angled Artin group and $S$ a subgroup of Thompson's group $V$ (resp. $T$, $F$). As an application, we show that several braided diagram groups embeds into Thompson's group $V$, including Higman's groups $V_{n,r}$, groups of quasi-automorphisms $QV_{n,r,p}$, and generalised Houghton's groups $H_{n,p}$.

math.GR↗

arXiv2019

Automorphisms of graph products of groups from a geometric perspective

This article studies automorphism groups of graph products of arbitrary groups. We completely characterise automorphisms that preserve the set of conjugacy classes of vertex groups as those automorphisms that can be decomposed as a product of certain elementary automorphisms (inner automorphisms, partial conjugations, automorphisms associated to symmetries of the underlying graph). This allows us to completely compute the automorphism group of certain graph products, for instance in the case where the underlying graph is finite, connected, leafless and of girth at least $5$. If in addition the underlying graph does not contain separating stars, we can understand the geometry of the automorphism groups of such graph products of groups further: we show that such automorphism groups do not satisfy Kazhdan's property (T) and are acylindrically hyperbolic. Applications to automorphism groups of graph products of finite groups are also included. The approach in this article is geometric and relies on the action of graph products of groups on certain complexes with a particularly rich combinatorial geometry. The first such complex is a particular Cayley graph of the graph product that has a quasi-median geometry, a combinatorial geometry reminiscent of (but more general than) CAT(0) cube complexes. The second (strongly related) complex used is the Davis complex of the graph product, a CAT(0) cube complex that also has a structure of right-angled building.

math.GR↗

arXiv2019

Rank-one isometries of CAT(0) cube complexes and their centralisers

If $G$ is a group acting geometrically on a CAT(0) cube complex $X$ and if $g \in G$ is an infinite-order element, we show that exactly one of the following situations occurs: (i) $g$ defines a rank-one isometry of $X$; (ii) the stable centraliser $SC_G(g)= \{ h \in G \mid \exists n \geq 1, [h,g^n]=1 \}$ of $g$ is not virtually cyclic; (iii) $\mathrm{Fix}_Y(g^n)$ is finite for every $n \geq 1$ and the sequence $(\mathrm{Fix}_Y(g^n))$ takes infinitely many values, where $Y$ is a cubical component of the Roller boundary of $X$ which contains an endpoint of an axis of $g$. We also show that (iii) cannot occur in several cases, providing a purely algebraic characterisation of rank-one isometries.

math.GR↗

arXiv2019

A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube complexes

In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group $F$, the lamplighter group $F \wr \mathbb{F}_2$ over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on a infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, roughly speaking, given a group $G$ acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup $L \leq G$, either the normaliser $N_G(L)$ is close to be free abelian or, for every $k \geq 1$, $N_G(L)$ contains a non-abelian free subgroup commuting with a subgroup of $L$ of size $\geq k$.

math.GR↗