Presentability by products for some classes of groups

TL;DR

This paper investigates which infinite groups can be described as quotients of products of two commuting infinite subgroups, with applications to various classes like 3-manifold groups and Coxeter groups.

Contribution

It characterizes groups presentable by products within certain classes and explores the structure of their factors, including finitely generated cases.

Findings

Identifies classes of groups presentable by products

Provides criteria for groups of small virtual cohomological dimension

Applies results to 3-manifold groups and Coxeter groups

Abstract

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.