Finite covers of random 3manifolds
Abstract
A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) > Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman's theorem that the action of the mapping class group on the SU(2) character variety is ergodic.
 Publication:

Inventiones Mathematicae
 Pub Date:
 July 2006
 DOI:
 10.1007/s0022200600016
 arXiv:
 arXiv:math/0502567
 Bibcode:
 2006InMat.166..457D
 Keywords:

 Mathematics  Geometric Topology;
 57M
 EPrint:
 60 pages