On the Tits alternative for $PD(3)$ groups

TL;DR

This paper proves the Tits alternative for a class of 3-dimensional Poincaré duality groups, showing that such groups either contain a free subgroup of rank 2 or are virtually cyclic, under certain conditions.

Contribution

It establishes the Tits alternative for almost coherent $PD(3)$ groups that are not virtually properly locally cyclic, a new result in geometric group theory.

Findings

Almost coherent $PD(3)$ groups contain a rank 2 free subgroup if not virtually cyclic.

Groups with fewer than four generators are excluded from containing free subgroups.

The Tits alternative holds for a broader class of 3-dimensional Poincaré duality groups.

Abstract

We prove the Tits alternative for an almost coherent $PD(3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD(3)$ group which cannot be generated by fewer than four elements always contains a rank 2 free group.