Crystallographic groups and flat manifolds from surface braid groups
| Autor: | Daciberg Lima Gonçalves, John Guaschi, Oscar Ocampo, Carolina de Miranda e Pereiro |
|---|---|
| Přispěvatelé: | Universidade de São Paulo (USP), Instituto de Matemática e Estatística (IME), Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Université de Caen Normandie (UNICAEN), Normandie Université (NU), Centre National de la Recherche Scientifique (CNRS), Universidade Federal da Bahia (UFBA), Universidade Federal do Espirito Santo (UFES) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Fundamental group
flat manifold Braid group Commutator subgroup Cyclic group Group Theory (math.GR) 01 natural sciences Anosov diffeomorphism [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] crystallographic group LAÇOS Mathematics::Group Theory Mathematics - Geometric Topology Conjugacy class Surface braid groups [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] FOS: Mathematics Order (group theory) Mathematics::Metric Geometry 0101 mathematics Frobenius group Mathematics 010102 general mathematics Geometric Topology (math.GT) Kähler manifold 010101 applied mathematics Crystallography Geometry and Topology Quotient group Mathematics - Group Theory |
| Zdroj: | Repositório Institucional da USP (Biblioteca Digital da Produção Intelectual) Universidade de São Paulo (USP) instacron:USP Journal of Algebra Journal of Algebra, Elsevier, 2021, 293, pp.107560. ⟨10.1016/j.topol.2020.107560⟩ |
| ISSN: | 0021-8693 1090-266X |
| DOI: | 10.1016/j.topol.2020.107560⟩ |
| Popis: | Let M be a compact surface without boundary, and n ≥ 2 . We analyse the quotient group B n ( M ) / Γ 2 ( P n ( M ) ) of the surface braid group B n ( M ) by the commutator subgroup Γ 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 ) / Γ 2 ( P n ( M ) ) ≅ P n ( M ) / Γ 2 ( P n ( M ) ) ⋊ φ S n , and that B n ( M ) / Γ 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 ) / Γ 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 ) / Γ 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 ) / Γ 2 ( P n ( M ) ) ⟶ S n is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups G ˜ n , g of B n ( M ) / Γ 2 ( P n ( M ) ) of dimension 2 n g and whose holonomy group is the finite cyclic group of order n, and if X n , g is a flat manifold whose fundamental group is G ˜ n , g , we prove that it is an orientable Kahler manifold that admits Anosov diffeomorphisms. |
| Databáze: | OpenAIRE |
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