We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that aren't quadratically hanging. Our main result is that any group quasi-isometric to $G$ is abstractly commensurable to $G$. In particular, our result applies to certain "generic" HNN extensions of a free group over cyclic subgroups.
We construct examples of groups showing that virtual solvability and the property of being virtually torsion-free are not preserved by bi-Lipschitz maps and hence by quasi-isometries.
We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We also show that Out(G) is virtually p-residually finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually p-residually finite.
We construct quasi-isometry invariants of a one-ended finitely presented group by considering the tree of cylinders of a two-ended JSJ decomposition of the group. When the group satisfies additional quasi-isometric rigidity hypotheses we construct finer invariants by also considering relative amounts of stretching across edges of the tree of cylinders.
We give a technical result that implies a straightforward necessary and sufficient conditions for a graph of groups with virtually cyclic edge groups to be one ended. For arbitrary graphs of groups, we show that if their fundamental group is not one-ended, then we can blow up vertex groups to graphs of groups with simpler vertex and edge groups. As an application, we generalize a theorem of Swarup to decompositions of virtually free groups.
In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph and we obtain new proofs of the mentioned results and other previously depending on them.
We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips classification of measured band complexes. Using canonical representatives, we deduce an algorithm for solving equations and inequations in hyperbolic groups (maybe with torsion). Additionnally, we can deal with quasi-isometrically embeddable rational constraints.
Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are always quasi-isometric, and a group G is quasi-isometrically rigid if every group quasi-isometric to G is virtually isomorphic to G. In this survey we describe quasi-isometric rigidity results for fundamental groups of manifolds which can be decomposed into geometric pieces. After stating by now classical results on lattices in semisimple Lie groups, we focus …
We show that if a 1-ended group $G$ acts geometrically on a CAT(0) space $X$ and $\bd X$ is separated by $m$ points then either $G$ is virtually a surface group or $G$ splits over a 2-ended group. In the course of the proof we study nesting actions on $\R $-trees and we show that nesting actions with non overlapping translation intervals give rise to isometric actions.
We present an algorithm that computes Bowditch's canonical JSJ decomposition of a given one-ended hyperbolic group over its virtually cyclic subgroups. The algorithm works by identifying topological features in the boundary of the group. As a corollary we also show how to compute the JSJ decomposition of such a group over its virtually cyclic subgroups with infinite centre. We also give a new algorithm that determines whether or not a given one-ended hyperbolic group is virtually fuchsian. Our approach uses only the geometry of large balls in the Cayley graph and avoids Makanin's algorithm.
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 amen…
We prove that if $T$ is an $\mathbb R$-tree with a minimal free isometric action of $F_N$, then the $Out(F_N)$-stabilizer of the projective class $[T]$ is virtually cyclic. For the special case where $T=T_+(ϕ)$ is the forward limit tree of an atoroidal iwip element $ϕ\in Out(F_N)$ this is a consequence of the results of Bestvina, Feighn and Handel, via very different methods. We also derive a new proof of the Tits alternative for subgroups of $Out(F_N)$ containing an iwip (not necessarily atoroidal): we prove that every such subgroup $G\le Out(F_N)$ is either virtually cyclic or contains a fre…
A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse $PD_n$ subgroup $H$ with $e(G/H)=\infty$, then whenever $G'$ is quasi-isometric to $G$, it contains an almost normal subgroup $H'$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G/H$ and $G'/H'$ are quasi-isometric. This generalises a theorem of Mosher-Sageev-Whyte, who prove the case in which $G/H$ is quasi-isometric to a fini…
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}(…
In this paper we provide a classification theorem for 1-dimensional boundaries of groups with isolated flats. Given a group $Γ$ acting geometrically on a $CAT(0)$ space $X$ with isolated flats and 1-dimensional boundary, we show that if $Γ$ does not split over a virtually cyclic subgroup, then $\partial X$ is homeomorphic to a circle, a Sierpinski carpet, or a Menger curve. This theorem generalizes a theorem of Kapovich-Kleiner, and resolves a question due to Kim Ruane. We also study the relationship between local cut points in $\partial X$ and splittings of $Γ$ over $2$-ended subgroups. In pa…
We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our result gives a new characterization of virtually fundamental groups of hyperbolic 3-manifolds. Keywords: CAT(0) cube complexes, Hyperbolic groups, hyperbolic 3-manifolds.
Profinite detection of free products and free factors
Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM（数学科学研究所）；Departamento de Matemáticas, Universidad Autónoma de Madrid（马德里自治大学数学系）；Departamento de Matemática, Universidade de Brasília（巴西利亚大学数学系）
We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admit a cocompact group action with quotient a tree. The conditions are stated in terms of two graphs we associate to the degree refinement of a group in this class. We prove there is a group in this class which is quasi-isometric to a Coxeter group but is not abstractly commensurabl…
We investigate the structure of the minimal displacement set in $8$-located complexes with the SD'-property. We show that such set embeds isometrically into the complex. Since $8$-location and simple connectivity imply Gromov hyperbolicity, the minimal displacement set in such complex is systolic. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on an $8$-located complex with the SD'-property.
Given a bounded valence, bushy tree T, we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T'. This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan's Theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first exam…
Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. In this paper, a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups, is shown to be closed under quasi-isometry. Additionally, these groups share some of the properties of limit groups. In particular, groups quasi-isometric to limit groups are shown to be LERF and virtually toral relatively hyperbolic.
We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL_n(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a h…
Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus g greater than 1 admits a (9g-8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean bui…
Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in $\mathcal{A}$. We show vertex finiteness when G is a toral relatively hyperbolic group and $\mathcal{A}$ is the family of abelian subgroups. We also show vertex finiteness when G is hyperbolic relative to virtually polycyclic subgroups and $\mathcal{A}$ is the family of virtually cyclic subgroups; if moreover G is one-ended, …
A monoid is said to be special if it admits a presentation in which all defining relations are of the form $w = 1$. Groups are familiar examples of special monoids. This article studies the geometric and structural properties of the Cayley graphs of finitely presented special monoids, building on work by Zhang and Gray-Steinberg. It is shown that the right Cayley graph $Γ$ of a special monoid $M$ is a context-free graph, in the sense of Muller & Schupp, if and only if the group of units of $M$ is virtually free. This generalises the geometric aspect of the well-known Muller-Schupp Theorem from…
We consider a finitely generated virtually abelian group $G$ acting properly and without inversions on a CAT(0) cube complex $X$. We prove that $G$ stabilizes a finite dimensional CAT(0) subcomplex $Y \subseteq X$ that is isometrically embedded in the combinatorial metric. Moreover, we show that $Y$ is a product of finitely many quasilines. The result represents a higher dimensional generalization of Haglund's axis theorem.
A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces an injective endomorphism phi of the Lie algebra g associated to the Lie group in which N is a lattice. A normal form for automorphisms of nilpotent Lie algebras--permuted absolute Jordan form-- is defined and conjectured to be a quasi-isometry invariant. We show that if phi, th…
We prove that if a surface group embeds as a normal subgroup in a Kähler group and the conjugation action of the Kähler group on the surface group preserves the conjugacy class of a non-trivial element, then the Kähler group is virtually given by a direct product, where one factor is a surface group. Moreover we prove that if a one-ended hyperbolic group with infinite outer automorphism group embeds as a normal subgroup in a Kähler group then it is virtually a surface group. More generally we give restrictions on normal subgroups of Kähler groups which are amalgamated products or HNN extension…
It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgako…
We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex-cocompact Kleinian group provided that its boundary has Ahlfors regular conformal dimension strictly less than $2$ or if it acts geometrically on a CAT(0) cube complex.
In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R …
We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S…
For a one-ended right-angled Artin group, we give an explicit description of its JSJ tree of cylinders over infinite cyclic subgroups in terms of its defining graph. This is then used to classify certain right-angled Artin groups up to quasi-isometry. In particular, we show that if two right-angled Artin groups are quasi-isometric, then their JSJ trees of cylinders are weakly equivalent. Although the converse to this is not generally true, we define quasi-isometry invariants known as stretch factors that can distinguish quasi-isometry classes of RAAGs with weakly equivalence JSJ trees of cylin…
We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group. In addition, we propose a candidate for a polycyclic group whose quasi-isometry group is a solvable real Lie group, and we introduce a candidate for a quasi-isometrically rigid solvable group that is not finitely presented. We also record some conjectures on the large…
In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably Z, then M does not cover any manifold (except itself). In the non-manifold case, Wright's method showed that when a one-ended, simply connected, locally compact ANR X with pro-monomorphic fundamental group at infinity admits an action of Z by covering transformations then the fundamental group at infinity of X is (up to pro-isomorphism) an inverse …
We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric prop…
