# Mapping classes are almost determined by their finite quotient actions

## TL;DR

This paper demonstrates that for surfaces, procongruent conjugacy classes of mapping classes are finite unions of conjugacy classes and many topological features are determined solely by these classes.

## Contribution

It establishes that procongruent conjugacy classes are finite unions of conjugacy classes and shows that key topological features depend only on these classes.

## Key findings

- Procongruent conjugacy classes are finite unions of conjugacy classes.
- Topological features like stretching factor and singularity type depend only on procongruent classes.
- Many invariants such as symplectic Floer homology are determined by procongruent conjugacy classes.

## Abstract

Given any connected compact orientable surface, a pair of mapping classes are said to be procongruently conjugate if they induce a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every characteristic finite quotient of the fundamental group. In this paper, it is shown that every procongruent conjugacy class of mapping classes, as a subset of the surface mapping class group, is the disjoint union of at most finitely many conjugacy classes of mapping classes. For any pseudo-Anosov mapping class of a connected closed orientable surface, several topological features are confirmed to depend only on the procongurent conjugacy class of the mapping class, including: the stretching factor, the topological type of the prong singularities, the transverse orientability of the invariant foliations, and the isomorphism type of the symplectic Floer homology.