Isoperimetric inequalities for Poincar\'e duality groups

TL;DR

This paper establishes that Poincaré duality groups over a ring are either amenable or satisfy a specific isoperimetric inequality, with applications including the Tits alternative and characterizations of surface groups.

Contribution

It proves a dichotomy for Poincaré duality groups over rings and provides new proofs and results regarding their structure and properties.

Findings

Groups are either amenable or satisfy a linear isoperimetric inequality.

The Tits alternative holds for 2-dimensional Poincaré duality groups.

When n=2 and R=Z, the group is a surface group.

Abstract

We show that every oriented $n$-dimensional Poincar\'e duality group over a $*$-ring $R$ is amenable or satisfies a linear homological isoperimetric inequality in dimension $n-1$. As an application, we prove the Tits alternative for such groups when $n=2$. We then deduce a new proof of the fact that when $n=2$ and $R = \mathbb Z$ then the group in question is a surface group.