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A note on $μ$-stabilizers in ACVF

Jinhe Ye

Abstract

We study $μ$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^μ(p)$ is an infinite unbounded definable subgroup of $G$ when $p$ is standard and unbounded. In the particular case when $G$ is linear algebraic, we show that $\mathrm{Stab}^μ(p)$ is a solvable algebraic subgroup of $G$, with $\mathrm{dim}(\mathrm{Stab}^μ(p))=\mathrm{dim}(p)$ when $p$ is $μ$-reduced and unbounded.

A note on $μ$-stabilizers in ACVF

Abstract

We study -stabilizers for groups definable in ACVF in the valued field sort. We prove that is an infinite unbounded definable subgroup of when is standard and unbounded. In the particular case when is linear algebraic, we show that is a solvable algebraic subgroup of , with when is -reduced and unbounded.

Paper Structure

This paper contains 8 sections, 13 theorems, 27 equations.

Table of Contents

  1. Acknowledgements.
  2. mu-types in ACVF
  3. Definable group actions in ACVF
  4. Computing mu-stabilizers via standard parts
  5. Tame topology on definable sets
  6. Proof of the main theorem
  7. Linear algebraic groups
  8. Examples

Key Result

Theorem 2

Let $G$ be a definable group in $\mathbf{VF}^n$ over an algebraically closed valued field $F$ considered as constants in the $\mathbf{VF}$-sort. Assume further that $G$ is closed with respect to the valuation topology and the group operations on $G$ are continuous. For a unbounded standard $G$-type

Theorems & Definitions (46)

  • Theorem 2
  • Remark 3
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.9
  • Proposition 1.10
  • proof
  • Definition 1.11
  • ...and 36 more