The action of mapping classes on nilpotent covers of surfaces

TL;DR

This paper investigates how infinite order mapping classes act on the homology of nilpotent covers of hyperbolic surfaces, revealing new properties of their actions and implications for the fundamental groups of associated 3--manifolds.

Contribution

It demonstrates that infinite order mapping classes induce infinite order actions on homology of certain nilpotent covers and explores the properties of Torelli classes and their suspensions.

Findings

Infinite order mapping classes act with infinite order on some nilpotent cover homology.

Torelli mapping classes either act infinitely on finite abelian cover homology or produce 3--manifolds with large fundamental groups.

Suspensions of Magnus kernel elements yield 3--manifolds with large fundamental groups.

Abstract

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.