Generalized Fibonacci groupsH(r,n,s) that are connected labelled oriented graph groups

Abstract

The class of connected LOG (Labelled Oriented Graph) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in S 4 , and so contains all knot groups. We investigate when Campbell and Robertson's generalized Fibonacci groups H(r, n, s) are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups H(r, n, s) that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that H(r, n, s) is a 2-generator knot group if and only if it is a torus knot group.

Perfect Prishchepov groups

“…Proof of Corollary B. To prove Corollary B, it suffices by Theorem A of [39] to show that if r ≡ 0 mod n and s ≡ 0 mod n, then the group H(r/2, n/2, s/2) is not perfect. Suppose for contradiction that these conditions hold but H(r/2, n/2, s/2) is perfect.…”

“…The groups H(r, n, s) = P (r, n, r + 1, s, 1) of [5], and the generalized Sieradski groups S(r, n) = P (r, n, 2, r − 1, 2), (r ≥ 2) of [8] play crucial roles in this work. In fact, the main motivation for working on this problem is the following conjecture by Williams (see [39]) regarding the perfectness of the groups H(r, n, s). Conjecture 1.2.…”

“…The main result of [39] is a partial classification of when the group H(r, n, s) is a connected Labelled Oriented Graph (LOG) group (see [39,24,22,23,25] for more details about LOG groups), the missing ingredient being Conjecture 1.2. Therefore, we are able to refine this result into a classification result as follows.…”

Cyclically presented groups as Labelled Oriented Graph groups

“…In Section 3 we consider the groups H(r, n, s). In Theorem 3.1 and Corollary 3.2 we obtain information about H(r, n, s) ab and in Corollary 3.3 we extend the classification of groups H(r, n, s) that are connected LOG groups [34,9] to classify all groups H(r, n, s) that are LOG groups. In Section 4 we turn our attention to the groups of Fibonacci type G n (m, k) and in Theorem 4.4 we show that if a group G n (m, k) is a LOG group then it is isomorphic to a Gilbert-Howie group H(n, m), and we consider these groups in Section 5.…”

“…Connections between HNN extensions of cyclically presented groups and LOG groups have been investigated in [12,15,29]. An almost complete classification of groups H(r, n, s) that are connected LOG groups was given in [34]; Chinyere and Bainson classify the perfect groups H(r, n, s) [9], completing the connected LOG groups classification. Asphericity of certain cyclic presentations of the form P n (x 0 wx −1 1 w −1 ) that are (connected) Word Labelled Oriented Graph presentations (or Wirtinger presentations) is established in [13,Section 3].…”