Crystallographic groups and flat manifolds from surface braid groups

  title={Crystallographic groups and flat manifolds from surface braid groups},
  author={Daciberg Lima Gonccalves and John Guaschi and Oscar Ocampo and Carolina de Miranda e Pereiro},
  journal={Topology and its Applications},
Abstract 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… Expand
Virtual braid groups, virtual twin groups and crystallographic groups
Let n ≥ 2. Let V Bn (resp. V Pn) be the virtual braid group (resp. the pure virtual braid group), and let V Tn (resp. PV Tn) be the virtual twin group (resp. the pure virtual twin group). Let Π beExpand


A quotient of the Artin braid groups related to crystallographic groups
Abstract Let n ≥ 3 . In this paper, we study the quotient group B n / [ P n , P n ] of the Artin braid group B n by the commutator subgroup of its pure Artin braid group P n . We show that B n / [ PExpand
On Bieberbach subgroups of B/[P,P] and flat manifolds with cyclic holonomy Z2d
Abstract Recently, Goncalves, Guaschi and Ocampo proved that, for n ≥ 3 , B n / [ P n , P n ] is a crystallographic group. Moreover, they showed that H ˜ = σ − 1 ( H ) [ P n , P n ] , where σ : B n →Expand
The inclusion of configuration spaces of surfaces in Cartesian products, its induced homomorphism, and the virtual cohomological dimension of the braid groups of S^2 and RP^2
Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F_n(M) --> M^n of the nthExpand
Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane
We consider the (pure) braid groups BnpMq and PnpMq, where M is the 2-sphere S2 or the real projective plane RP2. We determine the minimal cardinality of (normal) generating sets X of these groups,Expand
Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups.
Let $n\geq 3$. In this paper we show that for any finite abelian subgroup $G$ of $S_n$ the crystallographic group $B_n/[P_n,P_n]$ has Bieberbach subgroups $\Gamma_{G}$ with holonomy group $G$. UsingExpand
Almost-crystallographic groups as quotients of Artin braid groups
Let $n, k \geq 3$. In this paper, we analyse the quotient group $B\_n/\Gamma\_k(P\_n)$ of the Artin braid group $B\_n$ by the subgroup $\Gamma\_k(P\_n)$ belonging to the lower central series of theExpand
Embeddings of finite groups in B n /Γ k (P n ) for k=2,3
Let $n \geq 3$. In this paper, we study the problem of whether a given finite group $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k \inExpand
Anosov diffeomorphisms of flat manifolds
Let M be a compact differentiable manifold without boundary. A Riemannian structure on II is called flat if all sectional curvatures vanish at each point; then M is called a flat manifold AExpand
Geometry of Crystallographic Groups
  • A. Szczepanski
  • Mathematics, Computer Science
  • Algebra and Discrete Mathematics
  • 2012
The basic theory of crystallographic groups is developed from the very beginning, while in the second part, more advanced and more recent topics are discussed. Expand
Torsion subgroups of quasi-abelianized braid groups
This article extends the works of Goncalves, Guaschi, Ocampo [GGO] and Marin [MAR2] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group.Expand