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Camille Horbez

Publications and source records attributed to Camille Horbez.

At least 19 recordsLinked to original sources
arXiv2026

Algebraic and measurable embeddings between square-free right-angled Artin groups

Let $Γ,Δ$ be finite simplicial graphs, and assume that $Γ$ has no induced squares. We prove that the right-angled Artin group $A(Δ)$ measurably embeds into $A(Γ)$ if and only if $A(Δ)$ embeds as a subgroup in a graph product of free abelian groups over $Γ$. This in turn has a graph-theoretical characterisation which can be checked algorithmically. If additionally $A(Γ)$ and $A(Δ)$ have cohomological dimension equal to two and $A(Γ)$ is not isomorphic to $\mathbb{Z}\times F_n$, we get that $A(Δ)$ measurably embeds into $A(Γ)$ if and only if it embeds as a subgroup in $A(Γ)$. Our proof relies on the following algebraic statement. Let $K_1,\dots,K_n$ be a family of pairwise disjoint cliques in the square-free graph $Γ$ (or more generally in its extension graph $Γ^{\mathrm{ext}}$), and let $g_1,\dots,g_n$ be elements of $A(Γ)$ with respective parabolic supports $A(K_1),\dots,A(K_n)$. Then the subgroup of $A(Γ)$ generated by $g_1,\dots,g_n$ is a right-angled Artin group, where the only relations impose that $g_i$ and $g_j$ commute if $K_i\cup K_j$ is contained in a clique of $Γ$ (or of $Γ^{\mathrm{ext}}$).

math.GR↗
arXiv2026

Graph products and measure equivalence: classification, rigidity, and quantitative aspects

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no partial conjugation. This finds applications to their classification up to commensurability, and up to isomorphism, and to the study of their automorphism groups. We also derive structural properties of von Neumann algebras associated to probability measure-preserving actions of graph products. Variations of the measure equivalence classification statement are given with fewer assumptions on the defining graphs. We also provide a quantified version of our measure equivalence classification theorem, that keeps track of the integrability of associated cocycles. As an application, we solve an inverse problem in quantitative orbit equivalence for a large family of right-angled Artin groups. We then establish several rigidity theorems. First, in the spirit of work of Monod-Shalom, we achieve rigidity in orbit equivalence for probability measure-preserving actions of graph products, upon imposing extra ergodicity assumptions. Second, we establish a sufficient condition on the defining graph and on the vertex groups ensuring that a graph product G is rigid in measure equivalence among torsion-free groups (in the sense that every torsion-free countable group H which is measure equivalent to G, is in fact isomorphic to G). Using variations over the Higman groups as the vertex groups, we construct the first example of a group which is rigid in measure equivalence, but not in quasi-isometry, among torsion-free groups.

math.GR↗
arXiv2026

Isomorphisms and automorphisms of graph products of groups

We solve the isomorphism problem for graph products of groups. We give a generating set for the automorphism group of a graph product of groups, generalizing the one given by Laurence and Servatius for right-angled Artin groups.

math.GR↗
arXiv2025

PolExp growth for automorphisms of toral relatively hyperbolic groups

Let $G$ be a toral relatively hyperbolic group, and let $φ\in\mathrm{Aut}(G)$. We prove that, under iteration of $φ$, the conjugacy length $||φ^n(g)||$ of every element $g\in G$ grows like $n^dλ^n$ for some $d\in\mathbb{N}$ and some algebraic integer $λ\geq 1$. For a given $φ$, only finitely many values of $d$ and $λ$ occur as $g$ varies in $G$. The same statements hold for the growth of the word length $|φ^n(g)|$. For $G$ hyperbolic, we generalize polynomial subgroups: we show that, for a given growth type $n^dλ^n$ other than $1$, there is a malnormal family of quasiconvex subgroups $K_1,\dots,K_p$ such that a conjugacy class $[g]$ grows at most like $n^dλ^n$ if and only if $g$ is conjugate into one of the subgroups $K_i$.

math.GR↗
arXiv2025

Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry

Let $G$ be a right-angled Artin group with $|\mathrm{Out}(G)|<+\infty$. We prove that if a countable group $H$ with bounded torsion is measure equivalent to $G$, with an $L^1$-integrable measure equivalence cocycle towards $G$, then $H$ is finitely generated and quasi-isometric to $G$. In particular, through work of Kleiner and the second-named author, $H$ acts properly and cocompactly on a $\mathrm{CAT}(0)$ cube complex which is quasi-isometric to $G$ and equivariantly projects to the right-angled building of $G$. As a consequence of work of the second-named author, we derive a superrigidity theorem in integrable measure equivalence for an infinite class of right-angled Artin groups, including those whose defining graph is an $n$-gon with $n\ge 5$. In contrast, we also prove that if a right-angled Artin group $G$ with $|\mathrm{Out}(G)|<+\infty$ splits non-trivially as a product, then there does not exist any locally compact group which contains all groups $H$ that are $L^1$-measure equivalent to $G$ as lattices, even up to replacing $H$ by a finite-index subgroup and taking the quotient by a finite normal subgroup.

math.GR↗
arXiv2025

Rigidity for graph product von Neumann algebras

We establish rigidity theorems for graph product von Neumann algebras $M_Γ=*_{v,Γ}M_v$ associated to finite simple graphs $Γ$ and families of tracial von Neumann algebras $(M_v)_{v\inΓ}$. We consider the following three broad classes of vertex algebras: diffuse, diffuse amenable, and II$_1$ factors. In each of these three regimes, we exhibit a large class of graphs $Γ,Λ$ for which the following holds: any isomorphism $θ$ between $M_Γ$ and $N_Λ$ ensures the existence of a graph isomorphism $α:Γ\toΛ$, and tight relations between $θ(M_v)$ and $N_{α(v)}$ for every vertex $v\inΓ$, ranging from strong intertwining in both directions (in the sense of Popa), to unitary conjugacy in some cases. Our results lead to a wide range of applications to the classification of graph product von Neumann algebras and the calculation of their symmetry groups. First, we obtain general classification theorems for von Neumann algebras of right-angled Artin groups and of graph products of ICC groups. We also provide a new family of II$_1$ factors with trivial fundamental group, including all graph products of II$_1$ factors over graphs with girth at least $5$ and no vertices of degree $0$ or $1$. Finally, we compute the outer automorphism group of certain graph products of II$_1$ factors.

math.OA↗
arXiv2025

Measure equivalence rigidity of $\mathrm{Out}(F_N)$

We prove that for every $N\ge 3$, the group $\mathrm{Out}(F_N)$ of outer automorphisms of a free group of rank $N$ is superrigid from the point of view of measure equivalence: any countable group that is measure equivalent to $\mathrm{Out}(F_N)$, is in fact virtually isomorphic to $\mathrm{Out}(F_N)$. We introduce three new constructions of canonical splittings associated to a subgroup of $\mathrm{Out}(F_N)$ of independent interest. They encode respectively the collection of invariant free splittings, invariant cyclic splittings, and maximal invariant free factor systems. Our proof also relies on the following improvement of an amenability result by Bestvina and the authors: given a free factor system $\mathcal{F}$ of $F_N$, the action of $\mathrm{Out}(F_N,\mathcal{F})$ (the subgroup of $\mathrm{Out}(F_N)$ that preserves $\mathcal{F}$) on the space of relatively arational trees with amenable stabilizer is a Borel amenable action.

math.GR↗
arXiv2025

Measure equivalence rigidity of the handlebody groups

Let $V$ be a connected $3$-dimensional handlebody of finite genus at least $3$. We prove that the handlebody group $\mathrm{Mod}(V)$ is superrigid for measure equivalence, i.e. every countable group which is measure equivalent to $\mathrm{Mod}(V)$ is in fact virtually isomorphic to $\mathrm{Mod}(V)$. Applications include a rigidity theorem for lattice embeddings of $\mathrm{Mod}(V)$, an orbit equivalence rigidity theorem for free ergodic measure-preserving actions of $\mathrm{Mod}(V)$ on standard probability spaces, and a $W^*$-rigidity theorem among weakly compact group actions.

math.GR↗
arXiv2025

Measure equivalence rigidity among the Higman groups

We prove that all (generalized) Higman groups on at least $5$ generators are superrigid for measure equivalence. More precisely, let $k\ge 5$, and let $H$ be a group with generators $a_1,\dots,a_k$, and Baumslag-Solitar relations given by $a_ia_{i+1}^{m_i}a_i^{-1}=a_i^{n_i}$, with $i$ varying in $\mathbb{Z}/k\mathbb{Z}$ and nonzero integers $|m_i|\neq |n_i|$ for each $i$. We prove that every countable group which is measure equivalent to $H$, is in fact virtually isomorphic to $H$. A key ingredient in the proof is a general statement providing measured group theoretic invariants for groups acting acylindrically on $\mathrm{CAT}(-1)$ polyhedral complexes with control on vertex and edge stabilizers. Among consequences of our work, we obtain rigidity theorems for generalized Higman groups with respect to lattice embeddings and automorphisms of their Cayley graphs. We also derive an orbit equivalence and $W^*$-superrigidity theorem for all free, ergodic, probability measure-preserving actions of generalized Higman groups.

math.GR↗
arXiv2023

Rigidity of the Torelli subgroup in $Out(F_N)$

Let $N$ be at least 4. We prove that every injective homomorphism from the Torelli subgroup into $Out(F_N)$ differs from the inclusion by a conjugation in $Out(F_N)$. This applies more generally to the following subgroups: every finite-index subgroup of $Out(F_N)$ (recovering a theorem of Farb and Handel); every subgroup that contains a finite-index subgroup of one of the groups in the Andreadakis--Johnson filtration; every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich subgroup (subgroups that contain sufficiently many twists in an appropriate sense). Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in $Out(F_N)$; it also implies that all subgroups in the Andreadakis--Johnson filtration are co-Hopfian. We also prove the same rigidity statement for subgroups of $Out(F_3)$ which contain a power of every Nielsen transformation. This shows in particular that $Out(F_3)$ and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the $N=3$ case.

math.GR↗
arXiv2022

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_Γ\curvearrowright X$ and $G_Λ\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_Γ$ and $G_Λ$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_Γ\curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use works of Popa and Ioana-Popa-Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all ICC groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended non-abelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

math.GR↗
arXiv2022

Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$

We prove a rigidity result for cocycles from higher rank lattices to $\mathrm{Out}(F_N)$ and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let $G$ be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let $G\curvearrowright X$ be an ergodic measure-preserving action on a standard probability space, and let $H$ be a torsion-free hyperbolic group. We prove that every Borel cocycle $G\times X\to\mathrm{Out}(H)$ is cohomologous to a cocycle with values in a finite subgroup of $\mathrm{Out}(H)$. This provides a dynamical version of theorems of Farb--Kaimanovich--Masur and Bridson--Wade asserting that every morphism from $G$ to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image. The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.

math.GR↗
arXiv2022

On the homology growth and the $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$

Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ be the outer automorphism group of a free Coxeter group $W_n$ of rank $n$. We study the growth of the dimension of the homology groups (with coefficients in any field $\mathbb{K}$) along Farber sequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, in all degrees up to $\lfloor\frac{n}{2}\rfloor-1$, these Betti numbers grow sublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, through Lück's approximation theorem, this implies that all $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n}{2}\rfloor-1$. In contrast, in top dimension equal to $n-2$, an argument of Gaboriau and Noûs implies that the $\ell^2$-Betti number does not vanish. We also prove that the torsion growth of the integral homology is sublinear. Our proof of these results relies on a recent method introduced by Abért, Bergeron, Frączyk and Gaboriau. A key ingredient is to show that a version of the complex of partial bases of $W_n$ has the homotopy type of a bouquet of spheres of dimension $\lfloor\frac{n}{2}\rfloor-1$.

math.GR↗
arXiv2022

Measure equivalence classification of transvection-free right-angled Artin groups

We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification. However, in contrast with the quasi-isometry question, we observe that no right-angled Artin group is superrigid for measure equivalence in the strongest possible sense, for two reasons. First, a right-angled Artin group $G$ is always measure equivalent to any graph product of infinite countable amenable groups over the same defining graph. Second, when $G$ is nonabelian, the automorphism group of the universal cover of the Salvetti complex of $G$ always contains infinitely generated (non-uniform) lattices.

math.GR↗
arXiv2022

Proper proximality in non-positive curvature

Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces. First, these include many countable groups $G$ acting properly nonelementarily by isometries on a proper $\mathrm{CAT}(0)$ space $X$. More precisely, proper proximality holds in the presence of rank one isometries or when $X$ is a locally thick affine building with a minimal $G$-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary $\mathrm{CAT}(0)$ cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a $2$-dimensional piecewise Euclidean $\mathrm{CAT}(0)$ simplicial complex. Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph. In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions.

math.GR↗
arXiv2021

Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type

We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial isometries on a connected piecewise hyperbolic $\mathrm{CAT}(-1)$ simplicial complex with countably many simplices in finitely many isometry types, assuming that vertex stabilizers are boundary amenable. Consequently, they satisfy the Novikov conjecture. We then show that measure equivalent $2$-dimensional Artin groups of hyperbolic type have isomorphic fixed set graphs -- an analogue of the curve graph, introduced by Crisp. This yields classification results. We obtain strong rigidity theorems. Let $G=G_Γ$ be a $2$-dimensional Artin group of hyperbolic type, with $\mathrm{Out}(G)$ finite. When the automorphism groups of the fixed set graph and of the Cayley complex $\mathfrak{C}$ coincide, every countable group $H$ which is measure equivalent to $G$, is commensurable to a lattice in $\mathrm{Aut}(\mathfrak{C})$. This happens whenever $Γ$ is triangle-free with all labels at least $3$ -- unless $G$ is commensurable to the direct sum of $\mathbb{Z}$ and a free group. When $Γ$ satisfies an additional star-rigidity condition, then $\mathrm{Aut}(\mathfrak{C})$ is countable, and $H$ is almost isomorphic to $G$. This has applications to orbit equivalence rigidity, and rigidity results for von Neumann algebras associated to ergodic actions of Artin groups. We also derive a rigidity statement regarding possible lattice envelopes of certain Artin groups, and a cocycle superrigidity theorem from higher-rank lattices to $2$-dimensional Artin groups of hyperbolic type.

math.GR↗
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↗