A note on $\mu$-stabilizers in ACVF

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.