Title:
Finite quotients of 3-manifold groups
Abstract:
It is well-known that for any finite group G, there exists a closed
3-manifold M with G as a quotient of the fundamental group of M.
However, we can ask more detailed questions about the possible finite
quotients of 3-manifold groups, e.g. for G and H_1,...,H_n finite
groups, does there exist a 3-manifold group with G as a quotient but
no H_i as a quotient? We answer all such questions. To prove
non-existence, we prove new parity properties of the fundamental
groups of 3-manifolds. To prove existence of 3-manifolds with certain
finite quotients but not others, we use a probabilistic method, by
first proving a formula for the distribution of the fundamental group
of a random 3-manifold, in the sense of Dunfield-Thurston. This is
joint work with Will Sawin.