# Weak commutativity and finiteness properties of groups

## TL;DR

This paper investigates a group construction derived from a free product with added commutativity relations, revealing complex finiteness properties and homological behavior that depend on the original group's presentation.

## Contribution

It introduces a new group construction, analyzes its finiteness properties, and uncovers intricate homological features, especially for free groups, advancing understanding of group finiteness and homology.

## Key findings

- $rak{X}(G)$ is finitely presented iff $G$ is finitely presented
- For non-abelian free groups $F$, $rak{X}(F)$ has a subgroup with non-finitely generated third homology
- The construction yields finitely presented groups with complex homological properties

## Abstract

We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from this construction: $\mathfrak{X}(G)$ is finitely presented if and only if $G$ is finitely presented, but if $F$ is a non-abelian free group of finite rank then $\mathfrak{X}(F)$ has a subgroup of finite index whose third homology is not finitely generated.