On the 5/8 bound for non-Abelian Groups

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 $$ is satisfied. This resolves a conjecture by Langley, Levitt and Rower.