# Jonathan Hillman (University of Sydney)

## Friday 6th March, 12.05-12.55pm, Carslaw 375

### Indecomposable $PD_3$-complexes with virtually free fundamental group

In his 1997 thesis John Crisp showed that if a $PD_3$-complex $X$ with fundamental group $\pi$ is indecomposable then either the universal cover of $X$ is contractible or $\pi$ has a free subgroup of finite index. Several years ago in this seminar I described the first exotic" example: $\pi=S_3*_{Z/2Z}S_3$. In this talk we shall use one of Crisp's subsidiary results to determine the "generic" virtually free groups that arise in this way. (One case has resisted eviction or confirmation of tenure.) The methods are substantially group theoretic, and not particularly difficult (although I shall use the notion of graph of groups", which may be less familiar).