11/10/2005

Marc Lackenby
Oxford University

Counting subgroups and covering spaces in dimension 3

How many finite-sheeted covering spaces does a 3-manifold have, as a function of the covering degree? For hyperbolic 3-manifolds, the answer is not known, not even asymptotically. But it seems that if we are to make progress with important questions such as the Virtually Haken conjecture, then a good understanding of the`landscape' of all finite-sheeted covering spaces is required. In particular, we should know how many there are! In my talk, I will give some new lower bounds. For arithmetic 3-manifolds, these will be exponential in the covering degree. For general closed hyperbolic 3-manifolds, I will provide a slightly weaker lower bound, which uses a new theorem on the behaviour of mod p homology of finite index subgroups of a group.