A note on $\mu$-stabilizers in ACVF

TL;DR

This paper investigates $$-stabilizers in algebraically closed valued fields, revealing their structure as definable subgroups and their algebraic properties in specific cases, advancing understanding of model-theoretic stability in ACVF.

Contribution

It characterizes the structure of $$-stabilizers for definable groups in ACVF, especially relating to their algebraic and solvable nature in linear algebraic groups.

Findings

$$-stabilizers are infinite unbounded definable subgroups for standard, unbounded types.

In linear algebraic groups, $$-stabilizers are solvable algebraic subgroups.

Dimension of $$-stabilizers equals the dimension of the type when $$-reduced and unbounded.

Abstract

We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ 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 $\mathrm{Stab}^\mu(p)$ is a solvable algebraic subgroup of $G$, with $\mathrm{dim}(\mathrm{Stab}^\mu(p))=\mathrm{dim}(p)$ when $p$ is $\mu$-reduced and unbounded.