Cyclically presented Sieradski groups with even number of generators and three dimensional manifolds | Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 2019. № 60. DOI: 10.17223/19988621/60/3

Cyclically presented Sieradski groups with even number of generators and three dimensional manifolds

The Sieradski groups are defined by the presentation S(m)=〈x₁,x₂,...,x_m |x_ix_i₊₂ = x_i₊₁, i = 1,...,m〉 where all subscripts are taken by mod m . The generalized Sieradski groups S(m,p,q) are groups with m-cyclic presentation G_m(w) , where word w has a special form depending on coprime integers p and q . We study the problem if a given presentation is geometric, i.e. it corresponds to a spine of a closed orientable 3-manifold. It was shown by Cavicchioli, Hegenbarth, and Kim that the generalized Sieradski group presentation S(m,p,q) corresponds to a spine of some 3-manifold which we denote as M(m,p,q). Moreover, M(m,p,q) are m-fold cyclic coverings of S³ branched over the torus (p,q) -knot. Howie and Williams proved that M (2n,3,2) are n -fold cyclic coverings of the lens space L(3,1) . A. Vesnin and T. Kozlovskaya established that M (2n,5,2) are n-fold cyclic coverings of the lens space L(5,1) . In this paper, we consider generalized Sieradski manifolds M (2n,7,2) n ≥1. We prove that the n-cyclic presentations of their groups are geometric, i.e., correspond to spines of closed connected orientable 3-manifolds. Moreover, manifolds M (2n,7,2) are the n-fold cyclic coverings of the lens space L(7,1) . For the classification some of the constructed manifolds, we use the Recognizer computer program.

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Keywords

Sieradski group, cyclically presented group, lens space, branched covering, группа Сирадски, three-dimensional manifold, линзовое пространство, группа с циклическим представлением, разветвленное накрытие, трехмерное многообразие

Authors

Name	Organization	E-mail
Kozlovskaya Tatyana A.	Tomsk State University	t.kozlovskaya@math.tsu.ru

Всего: 1

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