Crystallographic groups and flat manifolds from surface braid groups

Gonçalves, Daciberg Lima; Guaschi, John; Ocampo, Oscar; Pereiro, Carolina de Miranda e

doi:10.1016/j.topol.2020.107560

Cited by 4 publications

(3 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the result below we exhibit explicit Bieberbach subgroups of V B n /Γ 2 (V P n ) with cyclic holonomy group. The proof of the next theorem is analogous to the one given for Theorem 3 of [14].…”

Section: Realization Of Bieberbach Subgroupsmentioning

confidence: 94%

“…Later, in [22], was studied the conjugacy problem and the realization of virtually cyclic groups in the group B n /Γ 2 (P n ). The surprising connection between Artin braid groups and crystallographic was extended to generalized braid groups associated to arbitrary complex reflection groups in [18], to orientable closed surfaces of positive genus in [14] and to punctured surfaces (orientable or not) in [11]. Was proved in [14,Proposition 17] that if M is the sphere or a closed non-orientable surface then, for every n ≥ 1, the quotient…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Virtual braid groups, virtual twin groups and crystallographic groups

Ocampo¹,

Júnior²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Let n ≥ 2. Let V B n (resp. V P n ) be the virtual braid group (resp. the pure virtual braid group), and let V T n (resp. P V T n ) be the virtual twin group (resp. the pure virtual twin group). Let Π be one of the following quotients:is the commutator subgroup of H. In this paper, we show that Π is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in Π. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in Π. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) module the respective commutator subgroup in each case.

show abstract

Section: Realization Of Bieberbach Subgroupsmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Virtual braid groups, virtual twin groups and crystallographic groups

Ocampo¹,

Júnior²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This quotient has been the subject of recent study, one of the reasons being that it is a crystallographic group [16][17][18]. The results of [16] have been generalised to quasi-Abelianised quotients of complex reflection groups [4,24], and to surface braid groups [19]. This enables us to give an alternative proof in Proposition 3.5 and Remark 3.6 of the fact that B n /[P n , P n ] embeds in the semi-direct product P n /[P n , P n ] ⋊ S n .…”

Section: Introductionmentioning

confidence: 99%