On some questions about a family of cyclically presented groups

Abstract

We study various questions about the generalised Fibonacci groups, a family of cyclically presented groups, which includes as special cases the Fibonacci, Sieradski, and Gilbert-Howie groups.

Largeness and Sq-Universality of Cyclically Presented Groups

“…Moreover the group H(9, 3) ∼ = H(9, 6) was proved to be infinite in [12,Lemma 15]. (We remark that the extension of this group also appears in [17, page 228] as G(−, 9).)…”

Fibonacci Type Semigroups

“…Except for two groups, this was provided in [13], [17], [18], [10]. The unresolved groups are the Gilbert-Howie groups ( [13]) H(9, 4) = R (2,9,6,4) and H(9, 7) = R(2, 9, 3, 7).…”

“…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).…”

Groups of Fibonacci Type Revisited

“…Question 2 of [1] asks if f (n) can be computed and this was investigated in [9]. Using the isomorphisms above, computer searches were used to obtain an upper bound U (n) on f (n); invariants of groups (usually the abelianization) were then used to obtain a lower bound L(n).…”

“…, x n−1 | x i x i+m = x i+k (0 ≤ i ≤ n − 1) (0 ≤ m, k ≤ n − 1) were introduced in [29]. Following the appearance of the groups G n (m, 1) in [21] in connection with Labelled Oriented Graph groups and the independent re-introduction of the groups G n (m, k) in [8] they have enjoyed renewed interest over the last decade [1], [9], [27], [46], [47], [48]. The groups G n (m, k) generalize various groups that have previously been studied: G n (1, 2) are Conway's Fibonacci groups F (2, n) of [12], the groups G n (2, 1) are the Sieradski groups S(2, n) of [41], and the groups G n (m, 1) are the Gilbert-Howie groups H(n, m) of [21].…”