Mathematics

This talk concerns groups $J_4(m,k)$ and $J_6(m,k)$ defined by generators and relations. The presentations arose in the study of asphericity for \textit{relative group presentations}, where one considers the effects of adding new generators and new relations to a pre-existing group. \textit{Asphericity} means that the pre-existing group embeds in the new group and that all finite subgroups of the new group are present in the old one, up to conjugacy. Non-asphericity entails the existence of certain planar diagrams, called \textit{pictures}, that represent essential spherical maps into cellular models for the group theory. The groups in question represent limiting cases where aspherical relative presentations transition to non-aspherical ones. In joint work with Gerald Williams (Essex) we prove that if $n = 4$ or $6$ and $m,k$ are relatively prime, then $J_n(m,k)$ are metabelian and finite with orders growing exponentially with $m$. This talk will focus on topological aspects of the work.