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Efficient Finite Groups Arising in the Study of Relative Asphericity

Bogley, WA and Williams, G (2016) 'Efficient Finite Groups Arising in the Study of Relative Asphericity.' Mathematische Zeitschrift, 284 (1). 507 - 535. ISSN 0025-5874

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Abstract

We study a class of two-generator two-relator groups, denoted Jn(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 Fa,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.

Item Type: Article
Uncontrolled Keywords: Relative presentation; asphericity; balanced presentation; deficiency zero; cyclically presented group; efficient group; Fibonacci group; resultant; Mersenne prime;Gaussian-Mersenne prime
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Jim Jamieson
Date Deposited: 29 Feb 2016 10:54
Last Modified: 07 Aug 2018 11:15
URI: http://repository.essex.ac.uk/id/eprint/16181

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