The commutator subgroups of free groups and surface groups

TL;DR

This paper provides a new geometric proof for Tomaszewski's free generating set of the commutator subgroup of free groups and extends the approach to surface groups, also analyzing their abelianizations and homology.

Contribution

It introduces a geometric proof for the generating set and generalizes it to surface groups, offering a representation-theoretic description of their abelianizations.

Findings

New geometric proof of Tomaszewski's theorem

Free generating set for surface group commutator subgroup

Calculation of homology and abelianization structure

Abstract

A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a surface group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.