Acylindrical hyperbolicity of groups acting on trees

  title={Acylindrical hyperbolicity of groups acting on trees},
  author={Ashot Minasyan and Denis V. Osin},
  journal={Mathematische Annalen},
We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, one-relator groups, automorphism groups of polynomial algebras, $$3$$3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes. 

Acylindrical group actions on quasi-trees

A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group G has a generating set X such that

On the acylindrical hyperbolicity of the tame automorphism group of SL 2(C)

We introduce the notion of über‐contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical

Orthogonal forms of Kac-Moody groups are acylindrically hyperbolic

We give sufficient conditions for a group acting on a geodesic metric space to be acylindrically hyperbolic and mention various applications to groups acting on CAT($0$) spaces. We prove that a group

Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles

We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with reduced irreducible classical root system of rank at least 2,

Acylindrical hyperbolicity of cubical small cancellation groups

We provide an analogue of Strebel's classification of geodesic triangles in classical $C'(\frac16)$ groups for groups given by Wise's cubical presentations satisfying sufficiently strong metric

Acylindrical hyperbolicity of the three-dimensional tame automorphism group

We prove that the group STame($k^3$) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of

Acylindrical hyperbolicity, non simplicity and SQ-universality of groups splitting over Z

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over Z cannot be simple. We also obtain SQuniversality in most cases, for instance a balanced group (one where if

A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes

We study the acylindrical hyperbolicity of groups acting by isometries on CAT(0) cube complexes, and obtain simple criteria formulated in terms of stabilisers for the action. Namely, we show that a


  • D. Osin
  • Mathematics
    Proceedings of the International Congress of Mathematicians (ICM 2018)
  • 2019
The goal of this article is to survey some recent developments in the study of groups acting on hyperbolic spaces. We focus on the class of acylindrically hyperbolic groups; it is broad enough to




ANDREAS THOMAbstract. We study the subgroup structure of discrete groups that sharecohomological properties which resemble non-negative curvature. Examplesinclude all Gromov hyperbolic groups.We

Acylindrical accessibility for groups

Abstract.We define the notion of acylindrical graph of groups of a group. We bound the combinatorics of these graphs of groups for f.g. freely indecomposable groups. Our arguments imply the

Normal subgroups in the Cremona group

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane $$ \mathbb{P}_{\mathbf{k}}^2 $$ is not a simple group. The strategy

Divergence and quasimorphisms of right-angled Artin groups

We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group AΓ with connected defining graph. We use this to prove that the

The virtual Haken conjecture

We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are

L2-Betti Numbers and Non-Unitarizable Groups without Free Subgroups

We show that there exist non-unitarizable groups without nonabelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely

Group actions on metric spaces: fixed points and free subgroups

We look at group actions on graphs and other metric spaces, e. g., at group actions on geodesic hyperbolic spaces. We classify the types of automorphisms on these spaces and prove several results

Profinite properties of graph manifolds

Let M be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π1(M) is efficient with respect to the JSJ decomposition of M.

The Structure of Groups with a Quasiconvex Hierarchy

Let $G$ be a word-hyperbolic group with a quasiconvex hierarchy. We show that $G$ has a finite index subgroup $G'$ that embeds as a quasiconvex subgroup of a right-angled Artin group. It follows

Conjugacy growth in polycyclic groups

In this paper, we consider the conjugacy growth function of a group, which counts the number of conjugacy classes which intersect a ball of radius n centered at the identity. We prove that in the