Wasserstein stability and the nonsingular Borel lifting problem

Abstract

The Borel lifting problem asks when a nonsingular near action of a Polish group can be represented by a genuine Borel action. For locally compact Polish groups, a classical theorem of Mackey and Ramsay gives an affirmative answer. At the opposite extreme, Glasner--Tsirelson--Weiss proved that every probability preserving Borel action of a L\'{e}vy group is trivial, and asked whether a L\'{e}vy group can admit a nontrivial nonsingular Borel action. We prove a fixed-point theorem which gives a negative answer for most of the standard L\'{e}vy groups in the literature: if $G$ is a locally Wasserstein group, then every quasi-invariant $\sigma$-finite Borel measure for a Borel action of $G$ is supported on the fixed points. The proof introduces Wasserstein stability, a local-to-global principle for compact measure metric groups that refines the Gromov--Milman framework for concentration of measure in topological groups. It asserts that for a compact measure metric group, on average, the total variation of locally distorted density kernels on the group controls their global Wasserstein distance from the Haar measure. We show that Wasserstein stability implies concentration, and prove a functional inequality using martingales in the spirit of Milman--Schechtman, which is governed by $\ell^{1}$-sums of martingale increments. This gives a geometric criterion by which most of the standard L\'{e}vy groups in the literature are Wasserstein or locally Wasserstein, including $L^{0}$-groups with compact targets, measure preserving and nonsingular automorphism groups, full groups of amenable equivalence relations, isometry groups of $L^{p}$-spaces for $p\neq2$, the unitary group of the hyperfinite $\mathrm{II}_{1}$-factor, the infinite dimensional unitary and orthogonal groups, the Cameron--Martin affine group, and the isometry group of the Urysohn space.