Virtual braid groups, virtual twin groups and crystallographic groups

Oscar Ocampo (Universidade Federal da Bahia)

Let . Let (resp. ) be the virtual braid group (resp. the pure virtual braid group), and let (resp. ) be the virtual twin group (resp. the pure virtual twin group). Let be one of the following quotients: or where is the commutator subgroup of .

In this talk, we show that 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 Júnior (arXiv: 2110.02392)

Slides for talk!