Linearly Bounded Conjugator Property for Mapping Class Groups

TL;DR

This paper proves that for conjugate mapping classes, a conjugating element can be found with length linearly bounded by the sum of the lengths of the classes, leading to an exponential bound on the conjugacy problem.

Contribution

It establishes a linearly bounded conjugator property for mapping class groups, providing a new bound on conjugator length relative to the classes involved.

Findings

Existence of a conjugator w with length < K(|f|+|g|)

Conjugacy problem for mapping class groups is exponentially bounded

Provides a new quantitative understanding of conjugacy in mapping class groups

Abstract

Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| < K(|f|+|g|), where |.| denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set. As a consequence, the conjugacy problem for mapping class groups is exponentially bounded.