Efficient Finite Groups Arising in the Study of Relative Asphericity

TL;DR

This paper investigates a special class of two-generator two-relator groups related to relative asphericity, revealing infinite families of virtually free and finite metabelian groups, and connecting their orders to prime number conjectures.

Contribution

It introduces and classifies the groups $J_n(m,k)$, extending existing group classes and linking their properties to prime number theory and cyclically presented groups.

Findings

Identifies infinite families of non-elementary virtually free groups.

Determines the orders of finite metabelian non-nilpotent groups.

Shows all Mersenne primes divide the orders of certain groups.

Abstract

We study a class of two-generator two-relator groups, denoted $J_n(m,k)$, that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature as finite groups of intriguing orders. Here we find infinite families of non-elementary virtually free groups and of finite metabelian non-nilpotent groups, for which we determine the orders. All Mersenne primes arise as factors of the orders of the non-metacyclic groups in the class, as do all primes from other conjecturally infinite families of primes. We classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups $F^{a,b,c}$ with trivalent Cayley graphs that was introduced by C.M.Campbell, H.S.M.Coxeter, and E.F.Robertson. The theory of cyclically presented groups informs our methods and we extend part of this theory (namely, on connections with polynomial resultants) to "bicyclically presented groups" that arise naturally in our analysis. As a corollary to our main results we obtain new infinite families of finite metacyclic generalized Fibonacci groups.