Virtual planar braid groups and permutations

TL;DR

This paper studies virtual twin groups, providing a complete description of homomorphisms to symmetric groups, and determines the automorphism group structure, revealing embeddings of twin groups and Coxeter groups within virtual twin groups.

Contribution

It offers a comprehensive analysis of homomorphisms and automorphisms of virtual twin groups, introducing the Coxeter group $KT_n$ and establishing embeddings of twin groups.

Findings

Explicit description of homomorphisms between virtual twin groups and symmetric groups.

Identification of the automorphism group structure of $VT_n$.

Embedding of twin groups into virtual twin groups.

Abstract

Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander-Markov correspondence for the theory of stable isotopy classes of immersed circles on orientable surfaces. Motivated by the general idea of Artin and a recent work of Bellingeri and Paris \cite{BellingeriParis2020}, we obtain a complete description of homomorphisms between virtual twin groups and symmetric groups, which as an application gives us the precise structure of the automorphism group of the virtual twin group $VT_n$ on $n \ge 2$ strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group $KT_n$ inside $VT_n$. As a by-product, it also follows that the twin group $T_n$ embeds inside the virtual twin group $VT_n$, which is an analogue of a similar result for braid groups.