Title: The co-rank of three-dimensional homology handlebody groups

Abstract: A group G is called very large if G has a non-abelian free quotient.  We examine the question of which three-manifolds have very large fundamental group.  This question is especially subtle for a three-dimensional homology handlebody of genus g, since the fundamental group of such a manifold is very close to being a free group of rank g.  We show that Thurston’s famous “tripus” manifold, a homology handlebody of genus 2, has fundamental group which is not very large.  It is still unknown whether every homology handlebody of genus at least 3 must have very large fundamental group.  However, we prove that for each g, there exists a homology handlebody of genus g whose fundamental group contains a free quotient of rank roughly half of g, but no greater.  This project was joint work with Shelly Harvey.