  The group of homeomorphisms of the closed interval that are orientation
preserving, absolutely continuous and have an absolutely continuous inverse was
shown by Solecki to admit a natural Polish group topology $\tau_{ac}$. We
observe that, more generally, under some conditions on a compact space endowed
with a finite Borel measure an analogous Polish group topology can be defined
on the subgroup of the homeomorphism group which push forward the measure to
another one with which it is mutually absolutely continuous. We use a
probabilistic argument involving approximations by supmartingale processes to
show that in many cases there is no group topology between $\tau_{ac}$ and the
restriction $\tau_{co}$ of the compact-open topology. This applies, in
particular, to any compact topological manifold equipped with an Oxtoby-Ulam
measure and to the Cantor space endowed with some natural Borel measures. In
fact, we show that in such cases any separable group topology strictly finer
than $\tau_{co}$ has to be also finer than $\tau_{ac}$. For one-dimensional
manifolds well-known arguments show that the compact-open topology is a minimum
Hausdroff group topology on the group, which implies that $\tau_{co}$ and
$\tau_{ac}$ are the only Hausdorff group topologies coarser than $\tau_{ac}$.
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.
