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Membership problems in braid groups and Artin groups

Robert D. Gray, Carl-Fredrik Nyberg-Brodda

TL;DR

The paper addresses the decidability of membership-type problems in braid groups and Artin groups. It combines embeddings of right-angled Artin groups into braid groups, notably showing that $A(P_4)$ embeds into $B_4$, with the Lohrey–Steinberg RAAG classification to establish a complete decidability picture for braids: $B_n$ has decidable submonoid membership and related problems precisely when $n\le 3$ (and undecidability occurs for $n\ge 4$). For general Artin groups, it provides a graph-theoretic dichotomy: submonoid and rational subset membership are decidable exactly when the defining graph $\Gamma$ avoids certain forbidden induced subgraphs; if such subgraphs occur, undecidability follows via embeddings of $A(P_4)$ or $A(C_4)$. The work further ties these results to subgroup separability, showing subgroup separable Artin groups lie in the $A(\mathcal{S})$ class and hence have decidable rational subset membership, and discusses outstanding questions about identity and group problems in Artin groups. Overall, the results unify algebraic and language-theoretic decision problems across braid and Artin groups and illuminate the sharp role of RAAG embeddings and graph structure in algorithmic behavior.

Abstract

We study several natural decision problems in braid groups and Artin groups. We classify the Artin groups with decidable submonoid membership problem in terms of the non-existence of certain forbidden induced subgraphs of the defining graph. Furthermore, we also classify the Artin groups for which the following problems are decidable: the rational subset membership problem, semigroup intersection problem, and the fixed-target submonoid membership problem. In the case of braid groups our results show that the submonoid membership problem, and each and every one of these problems, is decidable in the braid group $\mathbf{B}_n$ if and only if $n \leq 3$, which answers an open problem of Potapov (2013). Our results also generalize and extend results of Lohrey & Steinberg (2008) who classified right-angled Artin groups with decidable submonoid (and rational subset) membership problem.

Membership problems in braid groups and Artin groups

TL;DR

The paper addresses the decidability of membership-type problems in braid groups and Artin groups. It combines embeddings of right-angled Artin groups into braid groups, notably showing that embeds into , with the Lohrey–Steinberg RAAG classification to establish a complete decidability picture for braids: has decidable submonoid membership and related problems precisely when (and undecidability occurs for ). For general Artin groups, it provides a graph-theoretic dichotomy: submonoid and rational subset membership are decidable exactly when the defining graph avoids certain forbidden induced subgraphs; if such subgraphs occur, undecidability follows via embeddings of or . The work further ties these results to subgroup separability, showing subgroup separable Artin groups lie in the class and hence have decidable rational subset membership, and discusses outstanding questions about identity and group problems in Artin groups. Overall, the results unify algebraic and language-theoretic decision problems across braid and Artin groups and illuminate the sharp role of RAAG embeddings and graph structure in algorithmic behavior.

Abstract

We study several natural decision problems in braid groups and Artin groups. We classify the Artin groups with decidable submonoid membership problem in terms of the non-existence of certain forbidden induced subgraphs of the defining graph. Furthermore, we also classify the Artin groups for which the following problems are decidable: the rational subset membership problem, semigroup intersection problem, and the fixed-target submonoid membership problem. In the case of braid groups our results show that the submonoid membership problem, and each and every one of these problems, is decidable in the braid group if and only if , which answers an open problem of Potapov (2013). Our results also generalize and extend results of Lohrey & Steinberg (2008) who classified right-angled Artin groups with decidable submonoid (and rational subset) membership problem.
Paper Structure (10 sections, 13 theorems, 28 equations, 1 figure)

This paper contains 10 sections, 13 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Monoid presentations and rational subsets
  4. Membership problems
  5. Artin and braid groups
  6. Braid groups and right-angled Artin groups
  7. A connection between rational subsets and submonoids
  8. Further undecidability in $A(P_4)$
  9. Undecidability in the braid group $\mathbf{B}_4$
  10. Artin groups

Key Result

Theorem 3.1

The braid group $\mathbf{B}_n$ has decidable submonoid membership problem if and only $n \leq 3$. Furthermore, if $n \leq 3$ then $\mathbf{B}_n$ has decidable rational subset membership problem, while if $n \geq 4$ then $\mathbf{B}_n$ contains a fixed finitely generated submonoid in which membership

Figures (1)

  • Figure 1: The braid $\gamma_0$ from Theorem \ref{['Thm:Undecidability-b4']}.

Theorems & Definitions (25)

  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2: Lohrey & Steinberg Lohrey2008
  • Theorem 3.3: Droms, Lewin & Servatius Droms1991
  • proof : Proof of Theorem \ref{['thm:BraidClassification']}.
  • Lemma 3.4
  • proof
  • Claim
  • proof : Proof of Claim.
  • Lemma 3.5
  • ...and 15 more