Cyclically presented groups embedded in one-relator products of cyclic groups

Abstract

We consider the groups defined by the presentations \[ ⟨ a , b : a 2 = b n = a b − 1 a b ( a b a b − 1 ) α − 1 a b 2 a b − 2 = 1 ⟩ \langle a,b:{a^2} = {b^n} = a{b^{ - 1}}ab{(aba{b^{ - 1}})^{\alpha - 1}}a{b^2}a{b^{ - 2}} = 1\rangle \] and investigate their structure for small values of α \alpha . This forms part of a general investigation into the structure of groups defined by presentations of the form \[ ⟨ a , b : a 2 = b n = w ( a , b ) = 1 ⟩ . \langle a,b:{a^2} = {b^n} = w(a,b) = 1\rangle . \] Connections between these groups and the Fibonacci groups are also explored.