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A generalization of Fibonacci groups. (Russian, English) Zbl 1031.57001

Algebra Logika 42, No. 2, 131-160 (2003); translation in Algebra Logic 42, No. 2, 73-91 (2003).
The authors study a large class of groups with cyclic presentation \[ G_n(m,k)=G_n(x_1x_{1+m}x^{-1}_{1+k})=\langle x_1,\dots,x_n\mid x_ix_{i+m}=x_{i+k}, i=1,\dots,n\rangle. \] The groups \(G_n(m,k)\) turn into Fibonacci groups for \(m=1\), \(k=2\) and into Sieradski groups for \(m=2\), \(k=1\). The question is whether or not such groups are the fundamental groups of 3-manifolds.
Fibonacci groups and their generalisations were studied by a number of authors (see, for example, papers of R. Thomas, A. Yu. Vesnin, A. C. Kim, C. Maclachlan and A. W. Reid). In particular, in [C. Maclachlan, Combinatorial and geometric group theory. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 204, 233-238 (1995; Zbl 0851.20026)], it was proved that if \(n\) is even then a Fibonacci group is a fundamental group of a 3-manifold and moreover, for \(n\geq 8\), the 3-manifolds are hyperbolic. If, however, \(n\) is odd, then such groups are not realised as fundamental groups of 3-manifolds. Sieradski groups are always fundamental groups of 3-manifolds [A. Sieradski, Invent. Math. 84, 121-139 (1986; Zbl 0604.57001)].
The authors investigate the structure of the groups \(G_n(m,k)\). It is shown that under certain conditions \(G_n(m,k)\) is cyclic and can be decomposed as a free product. The question of group asphericity is a natural topological question arising here. Necessary conditions of asphericity of \(G_n(m,k)\) are given. It is also shown that \(G_n(m,k)\) such that \(n\) is odd, \(k\) is even, \((m-2k,n)=1\) are not realised as fundamental groups of finite volume hyperbolic 3-orbifolds.

MSC:

57M05 Fundamental group, presentations, free differential calculus
57M60 Group actions on manifolds and cell complexes in low dimensions
20F05 Generators, relations, and presentations of groups
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