The action of mapping classes on nilpotent covers of surfaces

Details The action of mapping classes on nilpotent covers of surfaces The action of mapping classes on nilpotent covers of surfaces

2011-10-17

arxiv.org/abs/1110.3743v1

Mathematics

Geometric Topology

18 pages

Scientific paper

Let $\Sigma$ be a surface whose interior admits a hyperbolic structure of finite volume. In this paper, we show that any infinite order mapping class acts with infinite order on the homology of some universal $k$--step nilpotent cover of $\Sigma$. We show that a Torelli mapping class either acts with infinite order on the homology of a finite abelian cover, or the suspension of the mapping class is a 3--manifold whose fundamental group has positive homology gradient. In the latter case, it follows that the suspended 3--manifold has a large fundamental group. It follows that every element of the Magnus kernel suspends to give a 3--manifold with a large fundamental group.

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