On One Application of Asphericity of Presentations
Maxim Zykin
TL;DR
This work addresses torsion in the group $\mathfrak{F}/[\mathfrak{M},\mathfrak{N}]$ arising from RC and combinatorially aspherical (CA) presentations. It provides a concise purely algebraic proof of Kulikova's Lemma 3(a) by leveraging the relation-module decomposition from Chiswell–Collins–Huebschmann, showing that $\mathfrak{N}/[\mathfrak{F},\mathfrak{N}]$ is a free abelian central subgroup with basis given by the images of the defining relators, which makes the extension $1 \to \mathfrak{N}/[\mathfrak{F},\mathfrak{N}] \to \mathfrak{F}/[\mathfrak{M},\mathfrak{N}] \to \mathfrak{G}$ torsion-free. The approach avoids geometric tools like spherical pictures and van Kampen diagrams, relying instead on an algebraic decomposition of the relation module. Consequently, the results imply that HNN-extensions and amalgamated free products built from CA presentations preserve CA and torsion-freeness, providing useful structural and cohomological consequences for groups presented with CA relations.
Abstract
I present a direct proof of Lemma 3(a) from O. V. Kulikova's work on torsion in the group $F/[M,N]$, using only Proposition 1.2 of Chiswell-Collins-Huebschmann on combinatorially aspherical presentations. In particular, I show that if two presentations satisfy the RC condition and are combinatorially aspherical, then the quotient $N/[F,N]$ is free abelian on the defining relators and central in $F/[F,N]$, whence the extension $F/[M,N]$ is torsion-free.