Group topologies on groups of bi-absolutely continuous homeomorphisms
J. de la Nuez González
TL;DR
The paper develops a comprehensive framework for bi-absolutely continuous homeomorphism groups $\mathcal{H}_{bac}(X,\mu)$, extending Solecki's Polish topology to OU pairs and the Cantor space. It constructs a Polish group topology $\tau_{ac}$ by combining the compact-open topology with an $L^{1}$-Jacobian metric $\rho_{ac}$ and proves a rigidity result: under mild assumptions there is no separable group topology strictly between the compact-open topology $\tau_{co}$ and $\tau_{ac}$. In one dimension, $\tau_{co}$ and $\tau_{ac}$ are the only Hausdorff topologies below $\tau_{ac}$, providing evidence against intermediate regularity notions between continuity and absolute continuity; and in the Cantor-Fräissé setting, the group is Roelcke precompact. The results illuminate how regularity between continuity and absolute continuity behaves for groups of homeomorphisms, and they separate different topological notions via fragmentation, measure-class preservation, and probabilistic methods.
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 $τ_{ac}$. 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 $τ_{ac}$ and the restriction $τ_{co}$ of the compact-open topology. In fact, we show that any separable group topology strictly finer than $τ_{co}$ must be also finer than $τ_{ac}$. For one-dimensional manifolds we also show that $τ_{co}$ and $τ_{ac}$ are the only Hausdorff group topologies coarser than $τ_{ac}$, and one can read our result as evidence for the non-existence 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 bi-absolutely continuous homeomorphisms of the Cantor space endowed with the measure given by the Fräissé limit of the class of measured boolean algebras with rational probability measures is Roelcke precompact.