Virtual braid groups, virtual twin groups and crystallographic groups

Oscar Ocampo (Universidade Federal da Bahia)

Let n\ge 2. Let VB_n (resp. VP_n) be the virtual braid group (resp. the pure virtual braid group), and let VT_n (resp. PVT_n) be the virtual twin group (resp. the pure virtual twin group). Let \Pi be one of the following quotients: VB_n/\Gamma_2(VP_n) or VT_n/\Gamma_2(PVT_n) where \Gamma_2(H) is the commutator subgroup of H.

In this talk, we show that \Pi is a crystallographic group and then and then we explore some of its properties, such as: characterization of finite order elements and its conjugacy classes, and also the realization of some Bieberbach groups and infinite virtually cyclic groups. Finally, we also consider other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) module the respective commutator subgroup in each case.

Joint work with Paulo Cesar Cerqueira dos Santos JĂșnior (arXiv: 2110.02392)

Slides for talk!

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