Peterzil-Steinhorn subgroups and $\mu$-stabilizers in ACF

TL;DR

This paper explores the structure of $ ext{Stab}^ ext{mu}$-types in algebraically closed fields, revealing solvable algebraic groups and linking their dimensions to types' dimensions.

Contribution

It introduces a new framework connecting $ ext{mu}$-types with algebraic group structures and describes their dimensions in algebraically closed valued fields.

Findings

$ ext{Stab}^ ext{mu}(p)$ is solvable and infinite for certain types.

$ ext{Stab}^ ext{mu}(p)$ can be described in terms of the dimension of $p$.

The paper links model-theoretic types to algebraic group properties.

Abstract

We consider $G$, a linear group defined over $k$, an algebraically closed field. By considering $k$ as an embedded residue field of an algebraically closed valued field $K$, we can associate to it a compact $G$-space $S^\mu_G(k)$, consisting of $\mu$-types on $G$. We showed that for each $p_\mu\in S^\mu_G(k)$, $\text{Stab}^\mu(p)=\text{Stab}(p_\mu)$ is a solvable infinite algebraic group when $p_\mu$ is centered at infinity and residually algebraic. Moreover we give a description of the dimension $\text{Stab}(p_\mu)$ in terms of dimension of $p$.