Group topologies on groups of absolutely bicontinuous homeomorphisms

J. de la Nuez González

Vol. 344 (2026), No. 1, 103–143

DOI: 10.2140/pjm.2026.344.103

Abstract

The group of homeomorphisms of the closed interval that are absolutely continuous and have an absolutely continuous inverse was shown by Solecki to admit a natural Polish group topology ${\tau }_{ac}$. We show that under mild conditions on a compact space endowed with a finite Borel measure such a topology can be defined on the subgroup of the homeomorphism group consisting of those elements $g$ such that $g$ and $g^{-1}$ preserve the class of null sets.

We use a probabilistic argument to show that in the case of a compact topological manifold equipped with an Oxtoby–Ulam measure, as well as in that of the Cantor space endowed with some natural Borel measures, there is no group topology between ${\tau }_{ac}$ and the restriction ${\tau }_{co}$ of the compact-open topology. In fact, we show that any separable group topology strictly finer than ${\tau }_{co}$ must be also finer than ${\tau }_{ac}$. For one-dimensional manifolds we also show that ${\tau }_{co}$ and ${\tau }_{ac}$ are the only Hausdorff group topologies coarser than ${\tau }_{ac}$, and one can read our result as evidence for the nonexistence of a good notion of regularity between continuity and absolute continuity.

We also show that while Solecki’s example is not Roelcke precompact, the group of absolutely bicontinuous homeomorphisms of the Cantor space endowed with the measure given by the Fraïssé limit of the class of measured boolean algebras with rational probability measures is Roelcke precompact. This can be interpreted as showing the nonexistence of a good notion of regularity between continuity and absolute continuity.

Keywords

group topology, absolutely continuous, homeomorphism group

Mathematical Subject Classification

Primary: 57S05, 57Sxx

Secondary: 28-XX

Milestones

Received: 6 October 2025

Revised: 26 February 2026

Accepted: 16 March 2026

Published: 10 August 2026