Constructing groups of type $FP_2$ over fields but not over the integers

TL;DR

This paper constructs specific groups that have finiteness properties over fields and finite fields but lack these properties over the integers, highlighting differences in group finiteness conditions across various rings.

Contribution

It provides explicit examples of groups with certain finiteness properties over fields and finite fields but not over the integers, illustrating nuanced distinctions in algebraic group theory.

Findings

Groups are $FP_2(Q)$ and $FP_2(Z/pZ)$ for all primes $p$

These groups are not of type $FP_2(Z)$

Demonstrates differences in finiteness properties over different rings

Abstract

We construct examples of groups that are $FP_2(\mathbb{Q})$ and $FP_2(\mathbb{Z}/p\mathbb{Z})$ for all primes $p$ but not of type $FP_2(\mathbb{Z})$.