Tadpole Labelled Oriented Graph groups and cyclically presented groups

Abstract

We study a class of Labelled Oriented Graph (LOG) group where the underlying graph is a tadpole graph. We show that such a group is the natural HNN extension of a cyclically presented group and investigate the relationship between the LOG group and the cyclically presented group. We relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the LOG group. In the case where the label on the tail of the LOG spells a positive word in the vertices in the circuit we show that the LOGs and groups coincide with those considered by Szczepański and Vesnin. We obtain new presentations for these cyclically presented groups and show that the groups of Fibonacci type introduced by Johnson and Mawdesley are of this form. These groups generalize the Fibonacci groups and the Sieradski groups and have been studied by various authors. We continue these investigations, using small cancellation and curvature methods to obtain results on hyperbolicity, automaticity, SQ-universality, and solvability of decision problems.

Groups of Fibonacci Type Revisited

“…In Corollary 8.2 we observe some consequences, most of which are well known properties of C(3)-T(6) and C(3)-T(7) groups (for details see the references in [27]). A countable group G is said to be SQ-universal if every countable group can be embedded in a quotient group of G; in particular, SQ-universal groups contain a free subgroup of rank 2.…”

“…We refer the reader to [32, Chapter V] for basic definitions regarding small cancellation theory. Using the characterization (from [23]) of the C(3)-T(q) conditions in terms of the star graph, the C(3)-T(6) and C(3)-T(7) presentations G n (m, k) were classified in [27]. (The term special means that all relators have length 3 -as is the case for G n (m, k) -and that the star graph is isomorphic to the incidence graph of a finite projective plane.…”

“…A countable group G is said to be SQ-universal if every countable group can be embedded in a quotient group of G; in particular, SQ-universal groups contain a free subgroup of rank 2. The dependence problems [37] DP(n), n ≥ 1, generalize the word problem (DP(1)) and the conjugacy problem (DP (2) 7] it can be shown (see [27]) that for odd primes p, G p (m, k) is either the trivial group, the Fibonacci group F (2, p), or the Sieradski group S(2, p) or is isomorphic to G p (1, (p + 1)/3) (when p ≡ 2 mod 3) or to G p (1, (p + 2)/3) (when p ≡ 1 mod 3 T(7). A group G n (m, k) may be hyperbolic and contain a free subgroup of rank 2 without its presentation G n (m, k) satisfying the C(3)-T(6) or C(3)-T(7) conditions; for example H(n, 3) is such a group for all n ≥ 11, n = 12, 14 ([27]).…”

“…, 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].…”

An Investigation Into the Cyclically Presented Groups with Length Three Positive Relators

“…Let n ≥ 3, 1 ≤ k, l < n, k = l and (n, k, l) = 1. If none of (B),(C),(D) hold then Γ n (k, l) contains a non-abelian free subgroup.We now record some other consequences which, as in the proof of[31, Corollary 11] (which concerns the groups G n (m, k)), follow from [36, Chapter V, Theorems 6.3 and 7.6], [22, Section 3]),[25, Theorem 2].…”