The Space of Actions, Partition Metric and Combinatorial Rigidity
Miklos Abert, Gabor Elek
TL;DR
The paper defines a partition metric $pd$ on the space $A(S)$ of probability-measure-preserving actions to crystallize Kechris’s weak equivalence as the zero set of a natural pseudometric, and proves that the resulting quotient is compact. Central to the approach are ultraproducts of probability spaces and of actions, together with a symbolic-dynamics reinterpretation via Schreier graphs and invariant random subgroups, which yields a robust local-to-global framework through global $k$-types $H^k_S(f)$. The authors show that two actions are weakly equivalent precisely when all corresponding $k$-types agree, and they establish compactness of the action space and of fibers over IRSs, enabling limit arguments and constructions of limit actions. A key application is that every free non-amenable action has a weakly equivalent action realizing the measurable von Neumann problem (via a Gaboriau–Lyons type embedding of a free $ ext{F}_2$ action), extended to unitary representations through the ultraproduct machinery. Altogether, the work provides a canonical, compactified moduli space for actions, connecting ergodic theory, representation theory, and combinatorial rigidity with clear consequences for measurable group theory.
Abstract
We introduce a natural pseudometric on the space of actions of d-generated groups. In this pseudometric, the zero classes correspond to the weak equivalence classes defined by Kechris, and the metric identification is compact. We achieve this by employing symbolic dynamics and an ultraproduct construction which also facilitates the extension of our results to unitary representations. As a byproduct, we show that the weak equivalence class of every free non-amenable action contains an action that satisfies the measurable von Neumann problem.