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Rotation sets and axes in the fine curve graph for torus homeomorphisms

Sebastian Hensel, Frédéric Le Roux

TL;DR

The paper establishes a deep link between rotation sets of torus homeomorphisms and their action on the fine curve graph, showing that axis data determines scaled rotation sets and that BF-equivalence forces translational relations among rotation sets. It introduces a metric WPD property for loxodromic elements in the torus setting, extends it via the fine marking graph and twist numbers, and derives impactful corollaries including a Tits-type alternative and positive stable commutator length in open sets of the homeomorphism group. The approach combines hyperbolic geometry, marking graphs, annular projections, and ping-pong arguments to connect dynamical invariants with algebraic properties, yielding new criteria for freeness, scl, and rotation-set stability, along with rich examples and counterexamples. Overall, the work provides a robust framework for translating axis-end data into rotation-set geometry, with broad implications for dynamics on the torus and the algebra of its homeomorphism group.

Abstract

We expand the dictionary between the action of a torus homeomorphism on the fine curve graph and its rotation set. More precisely, we show that the fixed points at infinity of a loxodromic element determine the rotation set up to scale. A key ingredient is a metric version of the classical WPD property from geometric group theory. As a consequence we find new stable criteria for positive scl, and for two homeomorphisms to generate a free group, and we provide a Tits alternative for groups of torus homeomorphisms.

Rotation sets and axes in the fine curve graph for torus homeomorphisms

TL;DR

The paper establishes a deep link between rotation sets of torus homeomorphisms and their action on the fine curve graph, showing that axis data determines scaled rotation sets and that BF-equivalence forces translational relations among rotation sets. It introduces a metric WPD property for loxodromic elements in the torus setting, extends it via the fine marking graph and twist numbers, and derives impactful corollaries including a Tits-type alternative and positive stable commutator length in open sets of the homeomorphism group. The approach combines hyperbolic geometry, marking graphs, annular projections, and ping-pong arguments to connect dynamical invariants with algebraic properties, yielding new criteria for freeness, scl, and rotation-set stability, along with rich examples and counterexamples. Overall, the work provides a robust framework for translating axis-end data into rotation-set geometry, with broad implications for dynamics on the torus and the algebra of its homeomorphism group.

Abstract

We expand the dictionary between the action of a torus homeomorphism on the fine curve graph and its rotation set. More precisely, we show that the fixed points at infinity of a loxodromic element determine the rotation set up to scale. A key ingredient is a metric version of the classical WPD property from geometric group theory. As a consequence we find new stable criteria for positive scl, and for two homeomorphisms to generate a free group, and we provide a Tits alternative for groups of torus homeomorphisms.

Paper Structure

This paper contains 46 sections, 82 theorems, 148 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Rotation sets and algebra
  3. Rotation sets and the fine curve graph
  4. The metric WPD property
  5. Ingredients for metric WPD
  6. Organization of the paper
  7. The rotation set
  8. Definition
  9. Rotation set bounds
  10. Hyperbolic preliminaries
  11. Basic definitions and the boundary
  12. Geodesics and quasigeodesics
  13. Isometry types and translation length
  14. Bestvina-Fujiwara equivalence relation
  15. Fine curve graphs
  16. ...and 31 more sections

Key Result

Theorem 1

Let $G$ be a subgroup of $\mathrm{Homeo}_0(T^2)$ which contains some element $f$ whose rotation set $\mathrm{Rot}(f)$ has interior. Then one of the following possibilities occurs:

Figures (18)

  • Figure 1: Lemma \ref{['lem:ftg']}
  • Figure 2: Lemma \ref{['lem:mid-quasigeodesics']}
  • Figure 3: Corollary \ref{['cor:midpoints']}
  • Figure 4: The motivating example for Lemma \ref{['lem:fellow-travel-tech']}
  • Figure 6: An elementary move of fine markings $\{\alpha, \beta\}$ to $\{\alpha, \beta'\}$
  • ...and 13 more figures

Theorems & Definitions (160)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1.1
  • proof : Proof of Corollary \ref{['coro:positive_scl_id']}
  • Theorem : Dagger2
  • Theorem 4
  • Theorem 5
  • proof : Theorem \ref{['thm:scl']} from Theorem \ref{['thm:equivalence']}
  • Definition 1.2
  • ...and 150 more