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Charmenability of arithmetic groups of product type

Abstract

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.

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Notes

  1. Beware that in some texts the term “character” is reserved for an extreme point in this set.

  2. See the discussion in p. 2 of [6] for some history of ideas.

  3. In fact the assumption that \(N^\Gamma = N^G\) would also imply the equivalence between (1) and (3).

  4. We emphasize that we are talking here about the support projections and not the central support projections.

  5. Which really is the algebra of all \(G_i\)-continuous elements in M.

  6. We note that the assumption that \(\mathbf{G}\) is simply connected is missing in this reference. This is certainly a typo, as this assumption is used in its proof. Of course when S is finite this doesn’t matter, but when S is infinite this assumption is necessary, as can be seen for example by the natural morphism \({\text {PGL}}_2(\mathbb {Q})\rightarrow \mathbb {Q}^*/(\mathbb {Q}^*)^2\) determined by the determinant morphism.

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Acknowledgements

We wish to thank Yair Glasner and Pierre-Emmanuel Caprace for providing us with the proof of Proposition 6.4. We thank Pierre-Emmanuel Caprace also for finding a flawed claim we made in an earlier version of this paper and for various suggestions for improvements of our presentation. We are grateful to Adrian Ioana and Narutaka Ozawa for their valuable remarks. Part of this work was done when CH was visiting the University of Tokyo during January-July, 2020. He would like to thank Yasuyuki Kawahigashi and the Graduate School of Mathematical Sciences for their kind hospitality. Last but not least, we thank the anonymous referees for carefully reading our paper and providing useful remarks.

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Correspondence to Rémi Boutonnet.

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UB is supported by ISF Moked 713510 Grant Number 2919/19.

RB is supported by a PEPS Grant from CNRS and ANR Grant AODynG, 19-CE40-0008

CH is supported by Institut Universitaire de France and FY2019 JSPS Invitational Fellowship for Research in Japan (long term)

JP is supported by NSF Grant DMS #1801125 and NSF FRG Grant #1853989.

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Bader, U., Boutonnet, R., Houdayer, C. et al. Charmenability of arithmetic groups of product type. Invent. math. (2022). https://doi.org/10.1007/s00222-022-01117-w

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Mathematics Subject Classification

  • 22D10
  • 22D25
  • 22E40
  • 37B05
  • 46L10
  • 46L30