Virtual braids and permutations

Paolo Bellingeri, Université de Caen Normandie. Part of the Algebra Seminar series.

Let VB_n the virtual braid group on $n$ strands and $S_n$ the symmetric group on $n$ elements. 
We determine all possible homomorphisms between:
- VB_n and VB_m 
- VB_n and  S_m
- S_n and VB_m
when n>4 and n\ge m. As corollaries we get several results on virtual braid groups, in particular we compute
their outer group and we show that virtual braid groups  are hopfian and co-hopfian. 
The approach is completely different from Artin and Lin ones
for classical braids and permutations, and it is based 
on Basse-Serre theory of amalgamated products of groups. This is a joint work with Luis Paris.