Obtaining presentations from group actions without making choices

TL;DR

This paper introduces a new method for deriving group presentations from actions on simplicial complexes that avoids arbitrary choices and is canonical, especially when the quotient space is 2-connected.

Contribution

It provides a novel, choice-free approach to obtain canonical presentations from group actions on simply-connected complexes, extending classical methods.

Findings

Method works without identifying a fundamental domain

Produces canonical group presentations

Applicable when the quotient is 2-connected

Abstract

Consider a group $G$ acting nicely on a simply-connected simplicial complex $X$. Numerous classical methods exist for using this group action to produce a presentation for $G$. For the case that $X/G$ is 2-connected, we give a new method that has the novelty that one does not have to identify a fundamental domain for the action. Indeed, the resulting presentation is canonical in the sense that no arbitrary choices need to be made. It can be viewed as a nonabelian analogue of a simple result in the study of equivariant homology.