Groups not presentable by products

TL;DR

This paper investigates obstructions to representing finitely generated groups as products, introducing acentral subgroups and linking algebraic properties with geometric and measure invariants, leading to new examples of non-presentable groups.

Contribution

It develops the concept of acentral subgroups and explores their relation to geometric and measure invariants, providing new examples of groups not presentable by products.

Findings

All groups with infinitely many ends are not presentable by products.

Automorphism groups of free groups are not presentable by products.

Some elementary amenable groups are not presentable by products.

Abstract

In this paper we study obstructions to presentability by products for finitely generated groups. Along the way we develop both the concept of acentral subgroups, and the relations between presentability by products on the one hand, and certain geometric and measure or orbit equivalence invariants of groups on the other. This leads to many new examples of groups not presentable by products, including all groups with infinitely many ends, the (outer) automorphism groups of free groups, Thompson's groups, and even some elementary amenable groups.