Crystallographic groups and flat manifolds from surface braid groups

Author(s)

Goncalves, Daciberg Lima ^[1] ; Guaschi, John ^[2] ; Ocampo, Oscar ^[3] ; de Miranda e Pereiro, Carolina ^[4]

Affiliation

^[1] IME USP, Dept Matemat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

^[2] Normandie Univ, Lab Math Nicolas Oresme UMR CNRS 6139, CNRS, UNICAEN, F-14000 Caen - France

^[3] Univ Fed Bahia, Dept Matemat, IME, Av Adhemar de Barros S-N, BR-40170110 Salvador, BA - Brazil

^[4] Univ Fed Espirito Santo, Dept Matemat, UFES, BR-29075910 Vitoria, ES - Brazil

• Document type: Journal article
• Source: Topology and its Applications; v. 293, APR 15 2021.
• Web of Science Citations: 0

Abstract

Let M be a compact surface without boundary, and n >= 2. We analyse the quotient group Bn(M)/Gamma(2)(P-n(M)) of the surface braid group B-n(M) by the commutator subgroup Gamma(2)(P-n(M)) of the pure braid group P-n(M). If M is different from the 2-sphere S-2, we prove that B-n(M)/Gamma(2)(P-n(M)) similar or equal to P-n(M)/Gamma(2)(P-n(M)) proportional to(phi) S-n, and that B-n(M)/Gamma(2)(P-n(M)) is a crystallographic group if and only if M is orientable. If M is orientable, we prove a number of results regarding the structure of B-n(M)/Gamma(2)(P-n(M)). We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of B-n(M)/Gamma(2)(P-n(M)) isomorphic either to S-n or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection B-n(M)/Gamma(2)(P-n(M)) -> S-n is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups (G) over tilde (n,g) of B-n(M)/Gamma(2)(P-n(M)) of dimension 2ng and whose holonomy group is the finite cyclic group of order n, and if X-n,X-g is a flat manifold whose fundamental group is (G) over tilde (n,g) we prove that it is an orientable Kahler manifold that admits Anosov diffeomorphisms. (C) 2020 Elsevier B.V. All rights reserved. (AU)

• FAPESP's process: 16/24707-4 - Algebraic, geometric and differential topology
• Grantee: Daciberg Lima Gonçalves
• Support Opportunities: Research Projects - Thematic Grants