Minimal proximal definable flows over the p-adics

Abstract

Let G be a definable group in an NIP theory. We prove that every minimal proximal definable G-flow is strongly proximal. Consequently, the universal minimal proximal definable G-flow Π^def(G) coincides with the minimal strongly proximal definable G-flow Π^def_s(G). Furthermore, for a p-adic definable group G, we can compute Π^def(G) explicitly. We show that Π^def(G) is exactly Π^def(S) where S is the semisimple part of the definably amenable-semisimple decomposition of G. In addition, Π^def(S)≅ S^*F(ℚ_p), the space of types of full dimension on F, where ℱ=F(ℚ_p) for a flag variety F constructed from S.

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BibTeX RIS

BibTeX

@misc{zhang2026minimal,
  title = {{Minimal proximal definable flows over the $p$-adics}},
  author = {Zhentao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03873},
  eprint = {2609.03873},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03873},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Minimal proximal definable flows over the p-adics
AU  - Zhentao Zhang
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.03873
UR  - https://arxiv.org/abs/2609.03873
AB  - Let G be a definable group in an NIP theory. We prove that every minimal proximal definable G-flow is strongly proximal. Consequently, the universal minimal proximal definable G-flow Π^def(G) coincides with the minimal strongly proximal definable G-flow Π^def_s(G). Furthermore, for a p-adic definable group G, we can compute Π^def(G) explicitly. We show that Π^def(G) is exactly Π^def(S) where S is the semisimple part of the definably amenable-semisimple decomposition of G. In addition, Π^def(S)≅ S^*_F(ℚ_p), the space of types of full dimension on F, where ℱ=F(ℚ_p) for a flag variety F constructed from S.
ER  -

Abstract

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