The aspherical Cavicchioli–Hegenbarth–Repovš generalized Fibonacci groups

Abstract

Abstract. The Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups are defined by the presentations G n ðm; kÞ ¼ hx 1 ; . . . ; x n j x i x iþm ¼ x iþk ð1 c i c nÞi. 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 G n ðm; kÞ. We determine when G n ðm; kÞ has infinite abelianization and provide su‰cient conditions for G n ðm; kÞ 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 CavicchioliHegenbarth-Repovš groups.

The cyclically presented groups with relators $x_ix_{i+k}x_{i+l}$

“…In connection with this and with Question 5 of [1] we note that this behaviour cannot occur for the groups H n .m; k/ (of the introduction) when they are finite (by [14], [15]), and that there are no recorded examples of it when they are infinite.…”

“…Using Theorem 2 of [14] and Theorem 3.2 of [10] we can obtain (the analogous result to Corollary 4.3) that the converse holds in many cases. For the interested reader we now state such a result where, for simplicity, we consider only the 'strongly irreducible' cases (see [14]). …”

“…It is hoped that the results here, together with those in [14], will provide insight into Problem 4.4 of [6], which asks for necessary and sufficient conditions for asphericity of those presentations.…”

“…We classify the finite groups G n .k; l/ and determine when the presentations P n .k; l/ are aspherical (that is, when 2 .K/ D 0 where K is the standard 2-dimensional CW-complex associated with P ). Similar investigations were carried out in [1], [14] for the cyclic presentations Q n .m; k/ with relators x i x iCm x 1 i Ck and the groups H n .m; k/ they define. (The groups H n .m; k/ were introduced in [4] and generalize Conway's Fibonacci groups F .2; n/ and the Sieradski groups S.2; n/).…”

Fibonacci Type Semigroups

“…More recently the groups R(2, n, k, h) -the so-called Cavicchioli-Hegenbarth-Repovš groups G n (h, h + k) -have been of interest for their algebraic and topological properties (see [3], [9]). With the exception of two unresolved cases the finite groups R(r, n, k, h) were classified in [17], [18], [10] and the present paper arose from a desire to classify the finite semigroups T (2, n, k, h). In doing so we found that the asphericity methods used in [3], [17] are effective in the more general setting and can be combined with the Adjan graph and semigroup rewriting techniques of [6], [7] to classify the finite semigroups T (r, n, k, h) in terms of the finite groups R(r, n, k, h).…”