Isoperimetric inequalities for Poincaré duality groups

Abstract

We show that every oriented $n$-dimensional Poincaré duality group over a $\ast$-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.

10 pages, 1 figure. v2: final part of the paper substantially expanded to accommodate referee's comments. To appear in Proc. Amer. Math. Soc