Topological Properties of Cyclically Presented Groups

Abstract

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

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

“…Moreover, as a corollary to Theorem 5.1 of [5] we know precisely when the abelianization is infinite. (Strictly, all parameters in that theorem are positive whereas we require one of them to be negative; this does not affect the proof in our case, however.)…”

Asphericity of symmetric presentations

“…Further results on the groups defined by these presentations and their generalizations can be found in [8].…”

Families of group presentations related to topology

“…(If n = 3, then F (2, n) is a finite group.) As special cases of Theorem 3.1 and Corollary 3.2, one can obtain the results on the non-hyperbolicity of certain groups of Fibonacci type proved in [1,6,15]. As a further result, we have The following arises in a natural way from the above results: = (a, b, r, s), m, k and h for which G ε n (m, k, h) is the fundamental group of closed connected orientable 3-manifolds for infinitely many n. Then classify the topological and geometric structures of such manifolds.…”

“…This class of groups was introduced in [5], and subsequently studied in [1] and [6]. It contains many well-known groups, e.g., the Fibonacci groups F (2, n), for m = 1 and k = 2, the Sieradski groups S(n), for m = 2 and k = 1, and the Gilbert-Howie groups, for k = 1 (see [8], [21] and [9], respectively).…”