On the 5/8 bound for non-Abelian Groups

TL;DR

This paper extends the known probability bound for commutativity in non-abelian groups to a topological setting, specifically for words representing fundamental groups of orientable surfaces, confirming a conjecture.

Contribution

It provides a topological generalization of the 5/8 bound for non-abelian groups, resolving a conjecture by Langley, Levitt, and Rower.

Findings

Established a probability estimate for words in surface groups

Resolved a conjecture on topological group properties

Extended the commutativity probability bound to surface group words

Abstract

If we pick two elements of a non-abelian group at random, the odds this pair commutes is at most 5/8, so there is a "gap" between abelian and non-abelian groups \cite{G}. We prove a "topological" generalization estimating the odds a word presenting the fundamental group of an orientable surface $<x,y: [x_1,y_1][x_2,y_2]...[x_n,y_n]=1>$ is satisfied. This resolves a conjecture by Langley, Levitt and Rower.