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The R$_{\infty}$-property for braid groups over orientable surfaces

Karel Dekimpe, Daciberg Lima Gonçalves, Oscar Ocampo

TL;DR

This work determines the $R_\infty$-property for surface pure braid groups $P_n(\Sigma_{g,p})$ and full surface braid groups $B_n(\Sigma_{g,p})$ on orientable finite-type surfaces, dividing the surfaces into three families and proving that, with a few small exceptions, these groups have infinitely many twisted conjugacy classes for every automorphism. It develops and leverages Goldberg's short exact sequence and the Fadell–Neuwirth sequence, together with automorphism identifications with extended mapping class groups, to transfer Reidemeister-number properties from quotients to the groups themselves. The main contributions include a complete treatment for the $\mathcal{F}_2$ family and substantial coverage for $\mathcal{F}_1$, yielding broad $R_\infty$-property results and clarifying exceptional low-complexity cases, thereby advancing Nielsen–Reidemeister theory for surface braid groups. The results have implications for fixed-point theory and twisted conjugacy phenomena in topological braid contexts, with potential applications to dynamics on configuration spaces of surfaces.

Abstract

Let $Σ_{g,p}$ be an orientable surface of genus $g$ and of finite type without boundary (i.e. an orientable closed surface with a finite number $p$ of points removed). In this paper we study the R$_{\infty}$-property for the surface pure braid groups $P_n(Σ_{g,p})$ as well as for the full surface braid groups $B_n(Σ_{g,p})$. We show that, with few exceptions, these groups have the R$_{\infty}$-property.

The R$_{\infty}$-property for braid groups over orientable surfaces

TL;DR

This work determines the -property for surface pure braid groups and full surface braid groups on orientable finite-type surfaces, dividing the surfaces into three families and proving that, with a few small exceptions, these groups have infinitely many twisted conjugacy classes for every automorphism. It develops and leverages Goldberg's short exact sequence and the Fadell–Neuwirth sequence, together with automorphism identifications with extended mapping class groups, to transfer Reidemeister-number properties from quotients to the groups themselves. The main contributions include a complete treatment for the family and substantial coverage for , yielding broad -property results and clarifying exceptional low-complexity cases, thereby advancing Nielsen–Reidemeister theory for surface braid groups. The results have implications for fixed-point theory and twisted conjugacy phenomena in topological braid contexts, with potential applications to dynamics on configuration spaces of surfaces.

Abstract

Let be an orientable surface of genus and of finite type without boundary (i.e. an orientable closed surface with a finite number of points removed). In this paper we study the R-property for the surface pure braid groups as well as for the full surface braid groups . We show that, with few exceptions, these groups have the R-property.

Paper Structure

This paper contains 10 sections, 7 theorems, 13 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Goldberg's short exact sequence for pure braid groups over punctured surfaces (orientable or not)
  3. The punctured sphere
  4. The punctured connected sum of tori
  5. The punctured non orientable surfaces
  6. Goldberg's short exact sequence for surface pure braid groups
  7. Automorphisms of $P_n(S)$ for $S$ a surface of finite type
  8. The R$_{\infty}$-property for $P_n(\Sigma_{g,p})$ and $B_n(\Sigma_{g,p})$
  9. The Fadell-Neuwirth short exact sequence
  10. The proof of Theorem \ref{['th:purerinfty']} and Theorem \ref{['th:totalrinfty']}

Key Result

Theorem 2

Let $\Sigma_{g,p}$ be a finite type surface which belongs to $\mathcal{F}_1\cup \mathcal{F}_2$. The surface pure braid group $P_n(\Sigma_{g,p})$ has the R$_{\infty}$-property if and only if one of the statements below holds:

Figures (4)

  • Figure 1: The punctured sphere.
  • Figure 2: Generator $A_{i,j}$ for $1\leq i \leq p+n-2$, $p\leq j\leq p+n-1$ and $i<j$.
  • Figure 3: Generator $A_{i,j}$ for $1\leq i\leq 2g+p+n-2$, $2g+p\leq j\leq 2g+p+n-1$ and $i<j$.
  • Figure 4: Generators $A_{i,j}$ and $\rho_{r,k}$ for $1\leq i<j$, $p+1\leq j,\, r\leq p+n$ and $1\leq k\leq g$.

Theorems & Definitions (19)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • proof
  • Lemma 9
  • ...and 9 more