Virtual planar braid groups and permutations

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 recent work of Bellingeri and Paris [P. Bellingeri and L. Paris, Virtual braids and permutations, Ann. Inst. Fourier (Grenoble) 70 (2020), 3, 1341–1362], 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

on

n ≥ 2 n\geq 2

strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group

KT n \mathrm{KT}_{n}

inside

VT n \mathrm{VT}_{n}

. As a by-product, it also follows that the twin group

T n \mathrm{T}_{n}

embeds inside the virtual twin group

VT n \mathrm{VT}_{n}

, which is an analogue of a similar result for braid groups.