Non-orientable surfaces have stably unbounded homeomorphism group

TL;DR

This paper proves that certain homeomorphism groups of non-orientable surfaces, like the real projective plane and Möbius strip, have elements with positive stable commutator length, addressing a key question in surface diffeomorphism theory.

Contribution

It demonstrates the existence of elements with positive stable commutator length in homeomorphism groups of specific non-orientable surfaces, completing previous inquiries about their boundedness.

Findings

Existence of homeomorphisms with positive stable commutator length

Addresses a question on boundedness of surface diffeomorphism groups

Utilizes recent results by Bowden, Hensel, and Webb

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 M\"obius 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.