Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Virtual braid groups, virtual twin groups and crystallographic groups   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 paper, we show that $\Pi$ is a
crystallographic group and we characterize the elements of finite order and the
conjugacy classes of elements in $\Pi$. Furthermore, we realize explicitly some
Bieberbach groups and infinite virtually cyclic groups in $\Pi$. 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.

27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an