Finite covers of random 3-manifolds

By Nathan M. Dunfield, William P. Thurston. 2005

148 citations recorded by the source corpus.

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds 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 3-manifold, 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 3-manifold 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 3-manifolds 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 led to consider the action of the mapping class group of a surface Σ on the set of quotients π1(Σ)→Q. If Q is a simple group, we show that if the genus of Σ 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/987743a674554ab25441be642d41f64708462409

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Citations in the corpus

A random tunnel number one 3–manifold does not fiber over the circle (2005, unknown): “For more on different possible models, and random 3–manifolds in general, see the work of the first author and W Thurston in [10].”
Large random simplicial complexes, I (2015, background): “ diﬀerent probabilistic models of random topological objects have appeared within the last 10 years, see [4] and [14] for surveys. One may mention random surfaces [18], random 3-dimensional manifolds [9], random conﬁguration spaces of linkages [11]. Linial, Meshulam and Wallach [15], [17] studied an”
Expander graphs in pure and applied mathematics (2011, background): “study the question whether the set of nite dimensional representations of the C-algebra C() of a nitely generated group, separate the points of C() (see [Be] and [LSh]). (VIII) Random 3-manifolds: In [DT], Duneld and Thurston presented a model for \random 3-manifolds". It is based on the fact (”
Topological methods in algebraic geometry (2015, methodology): “The above result extends a nice theorem of Livingston, Dunfield and Thurston ([47, 69]) in the unramifed case, where also the statement is simpler.”
The asymptotic growth of torsion homology for arithmetic groups (2010, background): “Since these groups are good, it would follow from Thurston’s conjecture (and the above two simple facts) that lattices in SL2(C) are good.”
Random walks on the mapping class group (2006, methodology): “gs give rise to hyperbolic manifolds, in analogy with Thurston’s hyperbolic Dehn surgery theorem. This was later stated speciﬁcally for random walks as part of Conjecture 2.10 in Dunﬁeld and Thurston [6], in which they also suggest that the volume of the hyperbolic manifold should grow linearly with”