Ergodic theory of affine isometric actions on Hilbert spaces

Yuki Arano, Yusuke Isono, Amine Marrakchi

Published 2019-11-11Version 1

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. Finally, we use Gaussian actions to show that every nonamenable locally compact group without property (T) admits a free nonamenable weakly mixing action of stable type III$_1$.