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Separability criteria for loops via the Goldman bracket

Aoi Wakuda

TL;DR

The paper develops algebraic separability criteria for free homotopy classes on oriented surfaces via the Goldman bracket, extending hyperbolic-geometric methods to loops that are not necessarily simple. It proves that for $m\ge 2$, $[x^{m},y]=0$ iff $i(x,y)=0$ or $y=x^{m}$, and establishes strong equivalences among multiple bracket-vanishing conditions equivalent to separability, all grounded in a detailed zigzag-curve analysis. A central result is the determination of the center of the Goldman Lie algebra for a pair of pants: it is generated by non-essential loops as a $K$-module, extending Kabiraj’s approach to this non-filled case. These contributions connect geometric intersection data with algebraic properties of the Goldman bracket, with potential implications for representations and skein-type constructions on surfaces.

Abstract

We provide some explicit algebraic criteria in terms of the Goldman bracket to decide whether two free homotopy classes of loops on an oriented surface admit disjoint representatives. We extend Kabiraj's method using the hyperbolic geometry of surfaces to prove these criteria. As an application, we show that the center of the Goldman Lie algebra of a pair of pants is generated by the class of the constant loop together with the classes of loops that wind multiple times around a single puncture or boundary component. This case was not covered by Kabiraj, since a pair of pants is not filled by simple closed curves.

Separability criteria for loops via the Goldman bracket

TL;DR

The paper develops algebraic separability criteria for free homotopy classes on oriented surfaces via the Goldman bracket, extending hyperbolic-geometric methods to loops that are not necessarily simple. It proves that for , iff or , and establishes strong equivalences among multiple bracket-vanishing conditions equivalent to separability, all grounded in a detailed zigzag-curve analysis. A central result is the determination of the center of the Goldman Lie algebra for a pair of pants: it is generated by non-essential loops as a -module, extending Kabiraj’s approach to this non-filled case. These contributions connect geometric intersection data with algebraic properties of the Goldman bracket, with potential implications for representations and skein-type constructions on surfaces.

Abstract

We provide some explicit algebraic criteria in terms of the Goldman bracket to decide whether two free homotopy classes of loops on an oriented surface admit disjoint representatives. We extend Kabiraj's method using the hyperbolic geometry of surfaces to prove these criteria. As an application, we show that the center of the Goldman Lie algebra of a pair of pants is generated by the class of the constant loop together with the classes of loops that wind multiple times around a single puncture or boundary component. This case was not covered by Kabiraj, since a pair of pants is not filled by simple closed curves.

Paper Structure

This paper contains 12 sections, 24 theorems, 31 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on the Goldman bracket and hyperbolic geometry
  3. The definition of the Goldman Lie algebra
  4. A generalized notion of intersection points
  5. Closed geodesics on a complete hyperbolic surface
  6. Hyperbolic isometries
  7. Separability criteria
  8. A zigzag curve induced by an intersection point of two closed geodesics
  9. Intersection of two zigzag curves
  10. Key results
  11. Proof of the separability criteria
  12. The center of the Goldman Lie algebra of a pair of pants

Key Result

Theorem 1.1

Goldman1986 Let $x, y \in \hat{\pi}$, where $x$ is represented by a simple closed curve. Then $[x, y] = 0$ in $\mathbb{Z}\hat{\pi}$ if and only if $i(x, y) = 0$.

Figures (15)

  • Figure 1: The forward angle $\phi_{P}(X)$
  • Figure 2: Relative position of the axes $A_g$, $A_h$, and $A_{gh}$ for hyperbolic isometries $g$, $h$, and $gh$.
  • Figure 3: A zigzag curve from alternating lifts of $\alpha(X)_P$ (blue) and $\beta(X)_P$ (red).
  • Figure 4: A zigzag curve $C$ and its reflection $D_u$ across $\mathcal{U}_u$, together with the geodesics $L$, $\mathcal{U}_u$, $\mathcal{V}_u$ and the points $M_i, N_i, P_i', P_i", Q_i', Q_i", H_i$.
  • Figure 5: Proof of Lemma \ref{['zigzag_lem1']} in the case $\theta_0 < \frac{\pi}{2}$
  • ...and 10 more figures

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['WSC2']}
  • Theorem 1.3: Theorem \ref{['SSC2']}
  • Theorem 1.4: Theorem \ref{['COPOP2']}
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 31 more