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PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF
Journal of the Institute of Mathematics of Jussieu  (IF1.522),  Pub Date : 2021-07-21, DOI: 10.1017/s147474802100030x
Moshe Kamensky, Sergei Starchenko, Jinhe Ye

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