Low degree bounded cohomology and l^2-invariants for negatively curved groups

TL;DR

This paper investigates the subgroup structures and homomorphic images of negatively curved groups, such as Gromov hyperbolic groups, revealing strong restrictions and properties related to their cohomological and geometric features.

Contribution

It provides new restrictions on s-normal subgroups, characterizes images of boundedly generated groups, and explores properties of groups admitting proper quasi-1-cocycles, extending to l2-orbit equivalence.

Findings

s-normal subgroups in hyperbolic groups are highly restricted

images of boundedly generated groups are virtually cyclic

groups with proper quasi-1-cocycles are closed under l2-orbit equivalence

Abstract

We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. We provide strong restrictions on the possible s-normal subgroups of a Gromov hyperbolic group, or more generally a 'negatively curved' group. Another result says that the image of a group, which is boundedly generated by a finite set of amenable subgroups, in a Gromov hyperbolic group has to be virtually cyclic. Moreover, we show that any homomorphic image of an analogue of a higher rank lattices in a Gromov hyperbolic group must be finite. These results extend to a certain class of randomorphisms in the sense of Monod. We study the class of groups which admit proper quasi-1-cocycles and show that it is closed under l2-orbit equivalence.