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Non-orientable surfaces have stably unbounded homeomorphism group

Lukas Böke

TL;DR

The paper resolves the question of whether the identity components of the homeomorphism groups of the real projective plane and the Möbius strip have unbounded stable commutator length, showing they are stably unbounded. It achieves this by leveraging Bavard duality to relate scl to homogeneous quasi-morphisms, and by proving hyperbolicity for the fine non-separating curve graphs via bicorn paths and WPD-type actions. The main technical engine is the adaptation of Bowden–Hensel–Webb techniques to non-orientable surfaces, including BSF structures and resolving coverings, to produce boundary points tied to ending foliations and to identify independent hyperbolic elements. These results complete the Burago–Ivanov–Polterovich program for surfaces (except the sphere) by establishing unbounded norms on Homeo_0 in the last remaining cases. The findings have a broad impact on understanding the geometry of surface homeomorphism groups and on the search for nontrivial quasi-morphisms in low-dimensional topology.

Abstract

Using a recent result of Bowden, Hensel and Webb, we prove the existence of homeomorphisms with positive stable commutator length in the groups of homeomorphisms of the real projective plane and Möbius strip which are isotopic to the identity. This completes the answer to a question posed by Burago, Ivanov and Polterovich on the boundedness of diffeomorphism groups of surfaces.

Non-orientable surfaces have stably unbounded homeomorphism group

TL;DR

The paper resolves the question of whether the identity components of the homeomorphism groups of the real projective plane and the Möbius strip have unbounded stable commutator length, showing they are stably unbounded. It achieves this by leveraging Bavard duality to relate scl to homogeneous quasi-morphisms, and by proving hyperbolicity for the fine non-separating curve graphs via bicorn paths and WPD-type actions. The main technical engine is the adaptation of Bowden–Hensel–Webb techniques to non-orientable surfaces, including BSF structures and resolving coverings, to produce boundary points tied to ending foliations and to identify independent hyperbolic elements. These results complete the Burago–Ivanov–Polterovich program for surfaces (except the sphere) by establishing unbounded norms on Homeo_0 in the last remaining cases. The findings have a broad impact on understanding the geometry of surface homeomorphism groups and on the search for nontrivial quasi-morphisms in low-dimensional topology.

Abstract

Using a recent result of Bowden, Hensel and Webb, we prove the existence of homeomorphisms with positive stable commutator length in the groups of homeomorphisms of the real projective plane and Möbius strip which are isotopic to the identity. This completes the answer to a question posed by Burago, Ivanov and Polterovich on the boundedness of diffeomorphism groups of surfaces.
Paper Structure (26 sections, 24 theorems, 21 equations)

This paper contains 26 sections, 24 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper.
  3. Acknowledgements.
  4. Background
  5. Groups of homeomorphisms
  6. Definitions.
  7. Point-pushing and surface braids.
  8. Pseudo-Anosov maps of non-orientable surfaces.
  9. Quasi-morphisms and commutator length
  10. Definitions.
  11. Bavard duality.
  12. Curve graphs and variations
  13. Hyperbolicity
  14. Actions on curve graphs
  15. BSF structures and resolving coverings
  16. ...and 11 more sections

Key Result

Theorem 1.1

The group of homeomorphisms of the real projective plane $\mathbb{R}\mathrm{P}^2$ and Möbius strip $M$ which are isotopic to the identity is stably unbounded with respect to commutator length in the sense of BIP.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 2.1: Bavard duality, bavard
  • Lemma 2.2: cf. Lemma 2.1 in masur-minsky and Lemma 4.3 in kuno-hyperbolicity
  • Lemma 2.3: cf. Lemma 6.4 of BHMMW
  • Definition 2.4
  • Theorem 2.5: Guessing geodesics
  • Definition 2.6
  • Proposition 2.7: Proposition 6 of Bestvina
  • Theorem 2.8: Theorem 1 of Bestvina
  • Definition 2.9
  • ...and 25 more