Efficient Finite Groups Arising in the Study of Relative Asphericity

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 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. Part of this research was carried out during a visit by the second named author to the Department of Mathematics at Oregon State University in July 2014. That visit was financed by a London Mathematical Society Scheme 4 grant (ref. 41332) and by the OSU College of Science and Department of Mathematics. The second named author would like to thank the LMS and OSU for its support and the OSU Department of Mathematics for its hospitality during that visit. William A.Bogley Department of Mathematics, Kidder Hall 368, Oregon State University, Corvallis, OR 973314605, USA. E-mail: [email protected] Gerald Williams Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, U.K. E-mail: [email protected] 2 William A. Bogley, Gerald Williams