# Acylindrical hyperbolicity of groups acting on trees

@article{Minasyan2013AcylindricalHO, title={Acylindrical hyperbolicity of groups acting on trees}, author={Ashot Minasyan and Denis V. Osin}, journal={Mathematische Annalen}, year={2013}, volume={362}, pages={1055-1105} }

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.

## 122 Citations

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## References

SHOWING 1-10 OF 95 REFERENCES

### LOW DEGREE BOUNDED COHOMOLOGY AND L 2 -INVARIANTS FOR NEGATIVELY CURVED GROUPS

- Mathematics
- 2007

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

- Mathematics
- 1997

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

- Mathematics
- 2010

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

- Mathematics
- 2010

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

- Mathematics
- 2012

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

- Mathematics
- 2008

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

- Mathematics
- 2013

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

- Mathematics
- 2008

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

- Mathematics
- 2009

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

- Mathematics
- 2010

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…