Topology and Geometric Group Theory Seminar
Sometimes a very nice group action on a space cannot be extended to an action of a slightly larger (finite index) supergroup. In this talk I'll discuss some simple examples of this phenomenon in the setting of groups that can or can't admit an action on a contractible 3-manifold. Kapovich-Kleiner have constructed a group G with the property that G is not the fundamental group of any 3-manifold, even though G has an index two subgroup that is a Kleinian group (acting on hyperbolic 3-space H^3). I will discuss a related family of groups formed as amalgams of surface groups over cyclic subgroups. These amalgams are all virtual 3-manifold groups, but many are not themselves 3-manifold groups.
I'll also discuss a related group H that is a 3-manifold group, but is not Kleininan, even though H has an index 2 Kleinian subgroup. This work is joint with Emily Stark and Hung Cong Tran.