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Virtual planar braid groups and permutations

Tushar Kanta Naik, Neha Nanda, Mahender Singh

TL;DR

The paper develops a structural framework for virtual twin groups VT$_n$ by isolating an irreducible right-angled Coxeter subgroup KT$_n$ inside VT$_n$ and exploiting the semidirect product VT$_n$ = KT$_n$ ⋊ S$_n$. KT$_n$ is presented as a RA Coxeter group generated by $\\alpha_{i,j}$ with $\alpha_{i,j}^2=1$ and disjoint commutativity, with S$_n$ acting transitively on these generators; this yields embeddings such as $T_n\hookrightarrow KT_n\hookrightarrow VT_n$ and enables a thorough analysis of homomorphisms VT$_n$ → S$_m$, S$_n$ → VT$_m$, and VT$_n$ → VT$_m$. The authors obtain complete classifications (up to conjugacy) of these homomorphisms in the stable range (n ≥ 5, m ≥ 2), reveal exceptional cases tied to the outer automorphism of S$_6$, and derive the automorphism structure Aut(VT$_n$) ≅ VT$_n$ ⋊ Z$_2$ for n ≥ 5 along with non-co-Hopficity. They also establish embedding results for the twin groups and provide detailed, technically robust lemmas via Reidemeister-Schreier analysis, amalgamated products, and fixed-point arguments to reduce general morphisms to canonical forms. The work thus connects planar and virtual doodle theories through precise group-theoretic descriptions, with implications for automorphisms and subgroup embeddings in VT$_n$ and its pure variants.

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.

Virtual planar braid groups and permutations

TL;DR

The paper develops a structural framework for virtual twin groups VT by isolating an irreducible right-angled Coxeter subgroup KT inside VT and exploiting the semidirect product VT = KT ⋊ S. KT is presented as a RA Coxeter group generated by with and disjoint commutativity, with S acting transitively on these generators; this yields embeddings such as and enables a thorough analysis of homomorphisms VT → S, S → VT, and VT → VT. The authors obtain complete classifications (up to conjugacy) of these homomorphisms in the stable range (n ≥ 5, m ≥ 2), reveal exceptional cases tied to the outer automorphism of S, and derive the automorphism structure Aut(VT) ≅ VT ⋊ Z for n ≥ 5 along with non-co-Hopficity. They also establish embedding results for the twin groups and provide detailed, technically robust lemmas via Reidemeister-Schreier analysis, amalgamated products, and fixed-point arguments to reduce general morphisms to canonical forms. The work thus connects planar and virtual doodle theories through precise group-theoretic descriptions, with implications for automorphisms and subgroup embeddings in VT and its pure variants.

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 on strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group inside . As a by-product, it also follows that the twin group embeds inside the virtual twin group , which is an analogue of a similar result for braid groups.

Paper Structure

This paper contains 10 sections, 36 theorems, 100 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Presentation of $KT_n$
  4. Homomorphisms from $VT_n$ to $S_m$
  5. Homomorphisms from $S_n$ to $VT_m$
  6. Technical results I
  7. Technical results II
  8. Technical results III
  9. Main results
  10. Homomorphisms from $VT_n$ to $VT_m$

Key Result

Theorem 1

For each $n \ge 2$, the group $KT_n$ is generated by $\mathcal{S}= \{ \alpha_{i, j} ~|~ 1 \leq i \neq j \leq n \}$, where $\alpha_{i, i+1}= s_i$ and $\alpha_{i+1, i}= \rho_i s_i \rho_i$. Further, the defining relations are the following:

Figures (4)

  • Figure 1: Real and virtual crossings
  • Figure 2: Reidemeister moves for virtual twin diagrams
  • Figure 3: Generator $s_i$ and $\rho_i$
  • Figure 4: Generators $\alpha_{i,j}$ and $\alpha_{j,i}$ of $KT_n$

Theorems & Definitions (62)

  • Theorem
  • Theorem
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Corollary 3.5
  • ...and 52 more