An Explicit Bound for the Conjugator Length Function of a Surface Group

TL;DR

This paper establishes explicit bounds for the conjugator length function in surface groups of genus at least 2, enhancing understanding of conjugation complexity in these fundamental groups.

Contribution

It provides the first explicit bounds for the conjugator length function in surface groups, derived through detailed conjugation reduction analysis.

Findings

Bounds: n-1 ≤ CL(2n) = CL(2n+1) ≤ n + 8g - 1

Improves understanding of conjugation complexity in surface groups

Offers tools for analyzing conjugation in geometric group theory

Abstract

For a surface group $\pi_1(\Sigma_g)=\langle c_1,\dots , c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\rangle$ with genus $g\geq 2$, we provide an explicit bound $n-1\leq \mathrm{CL}(2n)=\mathrm{CL}(2n+1)\leq n+8g-1$ for the conjugator length function $\mathrm{CL}:\mathbb N\to\mathbb N$ of $\pi_1(\Sigma_g)$ via a detailed analysis of conjugation reductions.