arXiv subjects
Publications and source records attributed to Camille Horbez.
We prove that $Out(F_N)$ is boundary amenable. This also holds more generally for $Out(G)$, where $G$ is either a toral relatively hyperbolic group or a finitely generated right-angled Artin group. As a consequence, all these groups satisfy the Novikov conjecture on higher signatures.
Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ is toral relatively hyperbolic). In the first case, either $H$ virtually preserves a nonperipheral conjugacy class in $G$, or else $H$ contains an atoroidal automorphism. The key geometric tool to obtain these classification results is a description of the Gromov boundaries of relative versions of the free factor graph $\mathrm{FF}$ and the $\mathcal{Z}$-factor graph $\mathcal{Z}\mathrm{F}$, as spaces of equivalence classes of arational trees (respectively relatively free arational trees). We also identify the loxodromic isometries of $\mathrm{FF}$ with the fully irreducible elements of $Out(G,\mathcal{F})$, and loxodromic isometries of $\mathcal{Z}\mathrm{F}$ with the fully irreducible atoroidal outer automorphisms.
We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $Σ$ be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that $\mathrm{Map}(Σ)$ admits a continuous nonelementary action on a hyperbolic space if and only if $Σ$ contains a finite-type subsurface which intersects all its homeomorphic translates. When $Σ$ contains such a nondisplaceable subsurface $K$ of finite type, the hyperbolic space we build is constructed from the curve graphs of $K$ and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. Our construction has several applications: first, the second bounded cohomology of $\mathrm{Map}(Σ)$ contains an embedded $\ell^1$; second, using work of Dahmani, Guirardel and Osin, we deduce that $\mathrm{Map}(Σ)$ contains nontrivial normal free subgroups (while it does not if $Σ$ has no nondisplaceable subsurface of finite type), has uncountably many quotients and is SQ-universal.
We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.
A theorem of Farb and Handel asserts that for $N\ge 4$, the natural inclusion from $\mathrm{Out}(F_N)$ into its abstract commensurator is an isomorphism. We give a new proof of their result, which enables us to generalize it to the case where $N=3$. More generally, we give sufficient conditions on a subgroup $Γ$ of $\mathrm{Out}(F_N)$ ensuring that its abstract commensurator $\mathrm{Comm}(Γ)$ is isomorphic to its relative commensurator in $\mathrm{Out}(F_N)$. In particular, we prove that the abstract commensurator of the Torelli subgroup $\mathrm{IA}_N$ for all $N\ge 3$, or more generally any term of the Andreadakis--Johnson filtration if $N\ge 4$, is equal to $\mathrm{Out}(F_N)$. Likewise, if $Γ$ the kernel of the natural map from $\mathrm{Out}(F_N)$ to the outer automorphism group of a free Burnside group of rank $N\geq 3$, then the natural map $\mathrm{Out}(F_N)\to\mathrm{Comm}(Γ)$ is an isomorphism.
This work is the first step towards a description of the Gromov boundary of the free factor graph of a free product, with applications to subgroup classification for outer automorphisms. We extend the theory of algebraic laminations dual to trees, as developed by Coulbois, Hilion, Lustig and Reynolds, to the context of free products; this also gives us an opportunity to give a unified account of this theory. We first show that any $\mathbb{R}$-tree with dense orbits in the boundary of the corresponding outer space can be reconstructed as a quotient of the boundary of the group by its dual lamination. We then describe the dual lamination in terms of a band complex on compact $\mathbb{R}$-trees (generalizing Coulbois-Hilion-Lustig's compact heart), and we analyze this band complex using versions of the Rips machine and of the Rauzy-Veech induction. An important output of the theory is that the above map from the boundary of the group to the $\mathbb{R}$-tree is 2-to-1 almost everywhere. A key point for our intended application is a unique duality result for arational trees. It says that if two trees have a leaf in common in their dual laminations, and if one of the trees is arational and relatively free, then they are equivariantly homeomorphic. This statement is an analogue of a result in the free group saying that if two trees are dual to a common current and one of the trees is free arational, then the two trees are equivariantly homeomorphic. However, we notice that in the setting of free products, the continuity of the pairing between trees and currents fails. For this reason, in all this paper, we work with laminations rather than with currents.
We define a new compactification of outer space $CV_N$ (the \emph{Pacman compactification}) which is an absolute retract, for which the boundary is a $Z$-set. The classical compactification $\overline{CV_N}$ made of very small $F_N$-actions on $\mathbb{R}$-trees, however, fails to be locally $4$-connected as soon as $N\ge 4$. The Pacman compactification is a blow-up of $\overline{CV_N}$, obtained by assigning an orientation to every arc with nontrivial stabilizer in the trees.
We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on $\text{Out}(F_N)$. In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time $n$ of the random walk to the Lyapunov exponent of the walk, which gives the typical growth rate of the length of a curve -- or of a conjugacy class in $F_N$ -- under a random product of diffeomorphisms/automorphisms. In the mapping class group case, we first observe that the drift of the random walk in the curve complex is also equal to the linear growth rate of the translation lengths in this complex. By using a contraction property of typical Teichmüller geodesics, we then lift the above fact to the realization of the random walk on the Teichmüller space. For the case of $\text{Out}(F_N)$, we follow the same procedure with the free factor complex in place of the curve complex, and the outer space in place of the Teichmüller space. A general criterion is given for making the lifting argument possible.
Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space $P\mathcal{O}(G,\{G_1,\dots,G_k\})$ for the axes topology with the space of projective minimal, \emph{very small} $(G,\{G_1,\dots,G_k\})$-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the $G_i$'s, and whose tripod stabilizers are trivial. Its topological dimension is equal to $3N+2k-4$, and the boundary has dimension $3N+2k-5$. We also prove that any very small $(G,\{G_1,\dots,G_k\})$-tree has at most $2N+2k-2$ orbits of branch points.
We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as $G=G_1\ast\dots\ast G_k\ast F$, where $F$ denotes a finitely generated free group, we identify the Gromov boundary of the graph of relative cyclic splittings with the space of equivalence classes of $\mathcal{Z}$-averse trees in the boundary of the corresponding outer space. A tree is \emph{$\mathcal{Z}$-averse} if it is not compatible with any tree $T'$, that is itself compatible with a relative cyclic splitting. Two $\mathcal{Z}$-averse trees are \emph{equivalent} if they are both compatible with a common tree in the boundary of the corresponding outer space. We give a similar description of the Gromov boundary of the graph of maximally-cyclic splittings.
We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on $\text{Out}(F_N)$, each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric on outer space). In the mapping class group case, this describes the spread of the hyperbolic length of a simple closed curve on the surface after applying a random product of mapping classes. In the case of $\text{Out}(F_N)$, this describes the spread of the length of primitive conjugacy classes in $F_N$ under random products of outer automorphisms. Both results are based on a general criterion for establishing a central limit theorem for the Busemann cocycle on the horoboundary of a metric space, applied to either the Teichmüller space of the surface, or to Culler--Vogtmann's outer space.
We prove that any isometry of the graph of cyclic splittings of a finitely generated free group $F_N$ of rank $N\ge 3$ is induced by an outer automorphism of $F_N$. The same statement also applies to the graphs of maximally-cyclic splittings, and of very small splittings.
Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in\{1,\dots,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy the Tits alternative, then $\text{Out}(G)$ satisfies the Tits alternative. As an application, we prove that the Tits alternative holds for outer automorphism groups of right-angled Artin groups, and of torsion-free groups that are hyperbolic relative to a finite family of virtually polycyclic groups.
We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions are homothetic in restriction to the set of primitive elements of $F_N$. We identify the set of Busemann points with the set of trees with dense orbits. We also investigate a few properties of the horoboundary of outer space for the backward Lipschitz metric, and show in particular that it is infinite-dimensional when $N\ge 3$. We then use our description of the horoboundary of outer space to derive an analogue of a theorem of Furstenberg--Kifer and Hennion for random products of outer automorphisms of $F_N$, that estimates possible growth rates of conjugacy classes of elements of $F_N$ under such products.
Let $μ$ be a probability measure on $\text{Out}(F_N)$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $\text{Out}(F_N)$. We show that almost every sample path of the random walk on $(\text{Out}(F_N),μ)$, when realized in Culler and Vogtmann's outer space, converges to the simplex of a free, arational tree. We then prove that the space $\mathcal{FI}$ of simplices of free and arational trees, equipped with the hitting measure, is the Poisson boundary of $(\text{Out}(F_N),μ)$. Using Bestvina-Reynolds' and Hamenstädt's description of the Gromov boundary of the complex $\mathcal{FF}_N$ of free factors of $F_N$, this gives a new proof of the fact, due to Calegari and Maher, that the realization in $\mathcal{FF_N}$ of almost every sample path of the random walk converges to a boundary point. We get in addition that $\partial\mathcal{FF}_N$, equipped with the hitting measure, is the Poisson boundary of $(\text{Out}(F_N),μ)$.
Two trees in the boundary of outer space are said to be \emph{primitive-equivalent} whenever their translation length functions are equal in restriction to the set of primitive elements of $F_N$. We give an explicit description of this equivalence relation, showing in particular that it is nontrivial. This question is motivated by our description of the horoboundary of outer space for the Lipschitz metric. Along the proof, we extend a theorem due to White about the Lipschitz metric on outer space to trees in the boundary, showing that the infimal Lipschitz constant of an $F_N$-equivariant map between the metric completion of any two minimal, very small $F_N$-trees is equal to the supremal ratio between the translation lengths of the elements of $F_N$ in these trees. We also provide approximation results for trees in the boundary of outer space.
We give a short proof of a theorem of Handel and Mosher stating that any finitely generated subgroup of $\text{Out}(F_N)$ either contains a fully irreducible automorphism, or virtually fixes the conjugacy class of a proper free factor of $F_N$, and we extend their result to non finitely generated subgroups of $\text{Out}(F_N)$.
Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's sphere complex (another model for the free splitting complex), instead of Handel and Mosher's fold paths. As a byproduct, we get that surgery paths are unparameterized quasi-geodesics in the sphere complex. We explain how to deduce from our proof the hyperbolicity of some other complexes such as the free factor complex or the arc complex (of a surface with boundary).