Subgroups Of Hyperbolic Groups Research Articles

Hyperbolicity and bounded-valued cohomology

AbstractWe generalise a theorem of Gersten on surjectivity of the restriction map in $$\ell ^{\infty }$$ ℓ ∞ -cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and $$\ell ^{\infty }$$ ℓ ∞ -cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type $$FP_2({{\mathbb {Q}}})$$ F P 2 ( Q ) and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Martínez-Pedroza.

Finitely generated groups acting uniformly properly on hyperbolic space

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ and not strongly contracting with respect to another. In the case of classical $C'(1/6)$ small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting. We show that many graphical $Gr'(1/6)$ small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups. In the course of our analysis we show that if the defining graph of a graphical $Gr'(1/6)$ small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.

The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology35 (2) 1999, 301–321).

Some prime factorization results for type II 1 factors

We prove several unique prime factorization results for tensor products of type II_1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if $R \otimes LF_{r_1} \otimes ... \otimes LF_{r_m}$ is isomorphic to a subfactor in $R \otimes LF_{s_1} \otimes >... \otimes LF_{s_n}$, for some $2\leq r_i, s_j \leq \infty$, then $m\le n$.

Branched Coverings of Cubical Complexes and Subgroups of Hyperbolic Groups

By considering branched coverings of piecewise Euclidean cubical complexes, the paper provides an example of a torsion free hyperbolic group containing a finitely presented subgroup which is not hyperbolic.