On quasihomomorphisms with noncommutative targets

TL;DR

This paper characterizes quasihomomorphisms from arbitrary groups to discrete groups, showing they are constructible from homomorphisms and simpler quasihomomorphisms, with specific results for hyperbolic groups.

Contribution

It introduces a structural description of quasihomomorphisms to discrete groups, demonstrating their constructibility from basic components and providing classifications for hyperbolic groups.

Findings

All quasihomomorphisms are constructible from homomorphisms and abelian quasihomomorphisms.

Unbounded quasihomomorphisms to torsion-free hyperbolic groups are either homomorphisms or to cyclic subgroups.

The structure theorem applies to various classes of groups, illustrating the generality of the approach.

Abstract

We describe structure of quasihomomorphisms from arbitrary groups to discrete groups. We show that all quasihomomorphisms are 'constructible', i.e., are obtained via certain natural operations from homomorphisms to some groups and quasihomomorphisms to abelian groups. We illustrate this theorem by describing quasihomomorphisms to certain classes of groups. For instance, every unbounded quasihomomorphism to a torsion-free hyperbolic group H is either a homomorphism to a subgroup of H or is a quasihomomorphism to an infinite cyclic subgroup of H.