On a Class of Finitely Presented Groups of Fibonacci Type

Largeness and Sq-Universality of Cyclically Presented Groups

“…(a) r = 2s, s = 2r and one of the following holds: 6 The finite Cavicchioli-Hegenbarth-Repovš groups…”

“…Examples of cyclic presentations of the trivial group are of interest in connection with Andrews-Curtis conjecture [1] and have been researched in [15], [24] and elsewhere. In contrast, papers such as [2], [6], [9], [16], [28], [33], [43] give conditions for a cyclically presented group to be infinite, and in [32] for it to be SQ-universal. The classification of finite cyclically presented groups within certain families is a problem addressed in, for example, [14], [23], [42], [45].…”

“…Thus there is the following almost complete classification of the finite groups H(n, t): Corollary 6.5 Suppose n ≥ 2, t ≥ 0, (n, t) = (9, 4), (9,7). Then H(n, t) is finite if and only if t = 0, 1 or (n, t) = (2k, k + 1) where k ≥ 1 (in which case H(n, t) ∼ = Z 2 k +1 ), or (n, t) = (3, 2), (4, 2), (5, 2), (5, 3), (5,4), (6,3), (7,4), (7,6), (8,3).…”

“…Suppose n ≥ 2, t ≥ 0, (n, t) = (8, 3), (9,3), (9,4), (9,6), (9,7) and suppose H(n, t) = 1. Then H(n, t) is finite if and only if t = 0, 1 or (n, t) = (2k, k + 1) where k ≥ 1 (in which case H(n, t) ∼ = Z 2 k +1 ), or (n, t) = (3, 2), (4, 2), (5, 2), (5, 3), (5,4), (6,3), (7,4), (7,6).…”

Asphericity of symmetric presentations

“…These groups were introduced in [4] as natural generalizations of the Fibonacci groups F (r, n). A lot of topological and algebraic results on these classes of groups can be found in the quoted paper and in [18].…”