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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On bounded elements of linear algebraic groups


Author: Kwan-Yuk Claire Sit
Journal: Trans. Amer. Math. Soc. 209 (1975), 185-198
MSC: Primary 22E20; Secondary 20G25
DOI: https://doi.org/10.1090/S0002-9947-1975-0379750-1
MathSciNet review: 0379750
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Abstract: Let $F$ be a local field of characteristic zero and ${\text {G }}$ a connected algebraic group defined over $F$. Let $G$ be the locally compact group of $F$-rational points. One characterizes the group $B(G)$ of $g \in G$ whose conjugacy class is relatively compact. For instance, if ${\text {G}}$ is $F$-split or reductive without anisotropic factors then $B(G)$ is the center of $G$. If $H$ is a closed subgroup of $G$ such that $G/H$ has finite volume, then the centralizer of $H$ in $G$ is contained in $B(G)$. If, moreover, $H$ is the centralizer of some $x \in G$ then $G/H$ is compact.


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Keywords: Local fields, automorphisms of bounded displacement, bounded elements, homogeneous spaces of finite volume, density theorem
Article copyright: © Copyright 1975 American Mathematical Society