Fibonacci type presentations and 3-manifolds

Abstract

We study the cyclic presentations with relators of the form xixi+mx −1 i+k and the groups they define. These "groups of Fibonacci type" were introduced by Johnson and Mawdesley and they generalize the Fibonacci groups F (2, n) and the Sieradski groups S(2, n). With the exception of two groups, we classify when these groups are fundamental groups of 3-manifolds, and it turns out that only Fibonacci, Sieradski, and cyclic groups arise. Using this classification, we completely classify the presentations that are spines of 3-manifolds, answering a question of Cavicchioli, Hegenbarth, and Repovš. When n is even the groups F (2, n), S(2, n) admit alternative cyclic presentations on n/2 generators. We show that these alternative presentations also arise as spines of 3-manifolds.

Aspherical Relative Presentations all Over Again

“…A three-manifold group is the fundamental group of a (PL) three-manifold, which need not be compact, closed, or orientable. A result of Ratcliffe [66] concerning rational Euler characteristics for groups [70] was exploited in [45] to establish a connection between relative asphericity and the three-manifold status for the group G(P). We make this connection explicit in the following result.…”

Hyperbolicity of $T$(6) cyclically presented groups

“…This family of groups contains the Fibonacci groups F (2, n) = G n (x 0 x 1 x −1 2 ) of [11], the Sieradski groups S(2, n) = G n (x 0 x 2 x −1 1 ) of [29], and the Gilbert-Howie groups H(n, m) = G n (x 0 x m x −1 1 ) of [17]. They have been subsequently studied in [1,30,8,20,21] -see [31] for a survey. In particular, the T(6) and T (7) presentations P n (x 0 x m x −1 k ) were classified in [20, Theorem 10] (see Corollary 3.2, below) and [20,Theorem 11] records that in the T (7) case the groups G n (x 0 x m x −1 k ) are non-elementary hyperbolic.…”

“…The free product of three copies of G 7 (x 0 x 1 x 5 ) is the cyclically presented group G 21 (x 0 x 3 x 15 ) with shift extension E = x, t | t 21 , xt 3 xt 12 xt −15 . The kernel of the retraction ν 7 : E → Z n = t | t21 given by ν 7 (t) = t, ν 7 (x) = t 7 is the group G 21 (x 0 x 10 x 8 ) which, by [14, Lemma 2.1(iv),(v)], is isomorphic to G 21 (x 0 x 1 x 5 ). Since G 7 (x 0 x 1 x 5 ) is not hyperbolic, neither is G 21 (x 0 x 3 x 15 ), nor E, and hence, nor is G 21 (x 0 x 1 x 5 ).…”

Hyperbolicity of T(6) Cyclically Presented Groups

“…This family of groups contains the Fibonacci groups F (2, n) = G n (x 0 x 1 x −1 2 ) of [11], the Sieradski groups S(2, n) = G n (x 0 x 2 x −1 1 ) of [28], and the Gilbert-Howie groups H(n, m) = G n (x 0 x m x −1 1 ) of [16]. They have been subsequently studied in [1], [29], [8], [19], [20] -see [30] for a survey. In particular, the T(6) and T (7) presentations P n (x 0 x m x −1 k ) were classified in [19, Theorem 10] (see Corollary 3.2, below) and [19,Theorem 11] records that in the T (7) case the group G n (x 0 x m x −1 k ) is non-elementary hyperbolic.…”