Cyclically presented groups with length four positive relators

Abstract

We classify cyclically presented groups of the form G = G n (x 0 x j x k x l ) for finiteness and, modulo two unresolved cases, we classify asphericity for the underlying presentations. We relate finiteness and asphericity to the dynamics of the shift action by the cyclic group of order n on the nonidentity elements of G and show that the fixed point subgroup of the shift is always finite.

Fractional Fibonacci groups with an odd number of generators

“…Theorem 5.1 is significant because it gives examples of infinite cyclically presented groups with torsion. Indeed, in many studies (for example [15,35,2,8,12,5,36]) cyclically presented groups are proved infinite by showing that they are non-trivial and that the relative presentations of their shift extensions are aspherical, and deducing (by [15,Lemma 3.1], [3, Theorem 4.1(a)]) that the cyclic presentation is topologically aspherical, and hence that the cyclically presented group is torsion-free.…”