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Strongly Choquet-Deny groups and invariant strictly convex renormings

  • Sheldon Dantas
  • Michal Doucha
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Abstract

We introduce and study a strengthening of the notion of Choquet-Deny groups, which we call \emph{strongly Choquet-Deny groups}. We show that a countable group G is strongly Choquet-Deny if and only if for every uniformly bounded representation of G on any separable Banach space X, there exists an equivalent G-invariant strictly convex norm on X. We provide a full classification of finitely generated strongly Choquet-Deny groups and show that a finitely generated group is strongly Choquet-Deny if and only if it is virtually abelian. This fully resolves a question of Mikael de la Salle for finitely generated groups. As a further application, we show that for every uniformly bounded representation of a compact extension of an abelian topological group on any separable Banach space, there exists an equivalent invariant strictly convex norm.

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Cite

BibTeXRIS
BibTeX
@misc{dantas2026strongly,
  title = {{Strongly Choquet-Deny groups and invariant strictly convex renormings}},
  author = {Sheldon Dantas and Michal Doucha},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23584},
  eprint = {2610.23584},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23584},
}
RIS
TY  - UNPB
TI  - Strongly Choquet-Deny groups and invariant strictly convex renormings
AU  - Sheldon Dantas
AU  - Michal Doucha
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23584
UR  - https://arxiv.org/abs/2610.23584
AB  - We introduce and study a strengthening of the notion of Choquet-Deny groups, which we call \emph{strongly Choquet-Deny groups}. We show that a countable group G is strongly Choquet-Deny if and only if for every uniformly bounded representation of G on any separable Banach space X, there exists an equivalent G-invariant strictly convex norm on X. We provide a full classification of finitely generated strongly Choquet-Deny groups and show that a finitely generated group is strongly Choquet-Deny if and only if it is virtually abelian. This fully resolves a question of Mikael de la Salle for finitely generated groups. As a further application, we show that for every uniformly bounded representation of a compact extension of an abelian topological group on any separable Banach space, there exists an equivalent invariant strictly convex norm.
ER  -