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PETERZIL–STEINHORN SUBGROUPS AND $\mu $-STABILIZERS IN ACF

Published online by Cambridge University Press:  21 July 2021

Moshe Kamensky
Affiliation:
Department of Mathematics, Ben-Gurion University, Be’er-Sheva, 8410501, Israel. (kamenskm@math.bgu.ac.il)
Sergei Starchenko
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, USA. (starchenko.1@nd.edu)
Jinhe Ye
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, 4, place Jussieu-Boite Courrier 247 75252 Paris Cedex 05, France. (jinhe.ye@imj-prg.fr)

Abstract

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.

Type
Research Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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