The aspherical Cavicchioli-Hegenbarth-Repovš Generalized Fibonacci groups

The Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups are defined by the presentationsGn(m, k) = 〈 x1, . . . , xn | xixi+m = xi+k (1 ≤ i ≤ n) 〉. These cyclically presented groups generalize Conway’s Fibonacci groups and the Sieradski groups. Building on a theorem of Bardakov and Vesnin we classify the aspherical presentations Gn(m, k). We determine when Gn(m, k) has infinite abelianization and provide sufficient conditions for Gn(m, k) to be perfect. We conjecture that these are also necessary conditions. Combined with our asphericity theorem, a proof of this conjecture would imply a classification of the finite Cavicchioli-Hegenbarth-Repovš groups.