Polyhedral products, and relations in the commutator subgroup of a right-angled Coxeter group

TL;DR

This paper characterizes when the commutator subgroup of a right-angled Coxeter group has a single defining relation, linking it to specific combinatorial structures of the underlying simplicial complex.

Contribution

It provides a precise criterion for the commutator subgroup to be a one-relator group based on the structure of the simplicial complex.

Findings

RC_{\ ext{K}}' is a one-relator group iff \mathscr{K} is a p-cycle with p≥4 or a (p-cycle)*\Delta^q with p≥4 and q≥0.

The criterion connects algebraic properties of the group with combinatorial features of the complex.

The result characterizes the algebraic structure of the commutator subgroup in terms of topological configurations.

Abstract

We give a criterion of the existence of a presentation with a single relation for the commutator subgroup $RC_{\mathscr{K}}'$ of a right-angled Coxeter group $RC_{\mathscr{K}}$. Namely, we prove that $RC_{\mathscr{K}}'$ is a one-relator group if and only if $\mathscr{K}$ is either a $p$-cycle with $p\geqslant4$ or has the form ($p$-cycle)$*\Delta^q$ with $p\geqslant4$ and $g\geqslant0$, where $\Delta^q$ is a $q$-simplex.