[Paper Review] Every Infinite order mapping class has an infinite order action on the homology of some finite cover

Asaf Hadari|arXiv (Cornell University)|Aug 6, 2015

Geometric and Algebraic Topology9 references7 citations

TL;DR

This paper proves that every infinite-order mapping class on a finite-type surface with nonabelian free fundamental group acts with infinite order on the first homology of some finite solvable cover. Using homological shadows, transition graphs, and Fourier analysis on abelian groups, the authors construct such covers explicitly, resolving a long-standing conjecture in mapping class group representation theory.

ABSTRACT

We prove the following well known conjecture: let $Σ$ be an oriented surface of finite type whose fundamental group is a nonabelian free group. Let $ϕ\in extup{Mod}(Σ)$ be a an infinite order mapping class. Then there exists a finite solvable cover $\widehatΣ o Σ$, and a lift $\widehatϕ$ of $ϕ$ such that the action of $\widehatϕ$ on $H_1(\widehatΣ, \mathbb{Z})$ has infinite order. Our main tools are the theory of homological shadows, which was previously developed by the author, and Fourier analysis

Motivation & Objective

• To resolve the well-known conjecture that every infinite-order mapping class on a punctured surface acts with infinite order on the homology of some finite cover.
• To establish a constructive method for building finite solvable covers where the lifted mapping class has infinite-order homological action.
• To extend results on homological representations of mapping class groups and automorphisms of free groups by proving the existence of such covers with controlled algebraic structure.
• To provide evidence toward McMullen's conjecture on the spectral radius of mapping class actions across finite covers.

Proposed method

• The authors use the theory of homological shadows to analyze the action of mapping classes on finite covers via transition graphs and associated matrices.
• They define extremal subgraphs and enfeoffed vertices to isolate components of the action with nontrivial spectral growth.
• Fourier analysis on finite abelian groups is applied to trace functions over homology, enabling the study of eigenvalue behavior across covers.
• The method involves lifting the mapping class to abelian covers and analyzing the trace of powers of associated matrices using character sums.
• By constructing a sequence of abelian covers and analyzing the image of the trace function under Fourier transforms, they detect nontrivial spectral growth.
• The key technical tool is the use of inverse Fourier transforms of roots of characteristic polynomials to detect non-vanishing trace terms over dense subsets of characters.

Experimental results

Research questions

• RQ1 Does every infinite-order mapping class on a surface with nonabelian free fundamental group act with infinite order on the first homology of some finite cover?
• RQ2 Can homological representations detect the infinite-order property of mapping classes through finite covers?
• RQ3 Is there a finite solvable cover such that the lifted mapping class has infinite-order action on homology, even when the original action is finite order?
• RQ4 Can the spectral radius of the homological action be made positive across some finite cover, as conjectured by McMullen?
• RQ5 Can the structure of the transition graph and its subgraphs be used to construct such covers algorithmically?

Key findings

• The paper proves that for any infinite-order mapping class on a surface with nonabelian free fundamental group, there exists a finite solvable cover where the lifted action on first homology has infinite order.
• The construction relies on identifying enfeoffed vertices in the transition graph and using abelian covers to amplify spectral growth in the homological representation.
• The authors show that if a subgraph is i-th level nilpotent enfeoffed, then there exists a cover where the trace of the k-th power of the monodromy matrix is nonzero for infinitely many k.
• The proof uses Fourier analysis on the dual group of an abelian group to detect nonvanishing contributions from eigenvalues off the unit circle.
• The method ensures that the resulting cover is solvable, satisfying a stronger condition than just being finite.
• The result confirms a long-standing conjecture and provides a constructive framework for detecting infinite-order behavior in homological representations.