Families of group presentations related to topology

Abstract

We study some algebraic properties of a class of group presentations depending on a finite number of integer parameters. This class contains many well-known groups which are interesting from a topological point of view. We find arithmetic conditions on the parameters under which the considered groups cannot be fundamental groups of hyperbolic 3-manifolds of finite volume. Then we investigate the asphericity for many presentations contained in our family.

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

“…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.…”

“…, (v) of Lemma 2.1 are contained in Lemmas 2.1 and 2.2 of [6]. More equivalences amongst the presentations P n .k; l/ can be established using the other results in Section 2 of [6].…”

“…(This is why we assume that .n; k; l/ D 1 in Theorem A.) Moreover, by Lemma 2.4 of [6], G n .k; l/ is isomorphic to the free product of d copies of the non-trivial group G n=d .k=d; l=d /, so G n .k; l/ is infinite when d > 1. Furthermore, if d D 1 and k D l then an elementary argument using Tietze transformations shows that G n .k; l/ Š Z s where s D 2 n .…”

“…More equivalences amongst the presentations P n .k; l/ can be established using the other results in Section 2 of [6].…”

“…The presentations P n .k; l/ and Q n .m; k/ fit into the more general class of cyclic presentations G .a;b;r;s/ n .m; k; h/ introduced by Cavicchioli, Repovš and Spaggiari in [6]. 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.…”