Group topologies on groups of bi-absolutely continuous homeomorphisms

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 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 Fr\"aiss\'e limit of the class of measured boolean algebras with rational probability measures is Roelcke precompact.