Abstract
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. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type \(\mathrm{III}_1\).
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Notes
The reason is that the Gaussian measure \(\mu _x(B)\) of the unit ball centered at x goes to 0 when the dimension of \({\mathcal {H}}\) goes to infinity
Recall that an orthogonal representation \(\pi : G \rightarrow {\mathcal {O}}(H)\) has stable spectral gap if the representation \(\pi \otimes \rho \) has no almost invariant vectors for every representation \(\rho : G \rightarrow {\mathcal {O}}(H)\).
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Acknowledgements
The authors express their deep gratitude to Stefaan Vaes for his interest in our work and for finding several mistakes in an earlier version of this paper. His advice also helped us to improve the presentation. We are grateful to Narutaka Ozawa for his numerous comments and for providing us with a correct proof of Proposition 4.11. We are grateful to Frederic Paulin for his explanations regarding the references [Su79] and [RT13]. We thank Michel Pain for explaining to us the additive martingale argument of Theorem 9.4. We thank Tushar Das for attracting our attention to the monograph [DSU17] and his comments on our paper. Finally we thank Amaury Freslon for pointing out to us the relations between our work and [Fr18] and [HP84].
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YI is supported by JSPS KAKENHI Grant Number JP18K13424.
YI is supported by JSPS KAKENHI Grant Number JP17K14201.
AM was a JSPS International Research Fellow (PE18760)
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Arano, Y., Isono, Y. & Marrakchi, A. Ergodic theory of affine isometric actions on Hilbert spaces. Geom. Funct. Anal. 31, 1013–1094 (2021). https://doi.org/10.1007/s00039-021-00584-2
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DOI: https://doi.org/10.1007/s00039-021-00584-2
Keywords
- Affine isometric action
- Hilbert space
- Gaussian measure, nonsingular action
- Type III
- Phase transition
- Trees
- Orthogonal representation
- Property (T)