## On bounded elements of linear algebraic groups

HTML articles powered by AMS MathViewer

- by Kwan-Yuk Claire Sit PDF
- Trans. Amer. Math. Soc.
**209**(1975), 185-198 Request permission

## 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.## References

- Armand Borel,
*Density properties for certain subgroups of semi-simple groups without compact components*, Ann. of Math. (2)**72**(1960), 179–188. MR**123639**, DOI 10.2307/1970150 - Armand Borel,
*Linear algebraic groups*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR**0251042** - Armand Borel and Jacques Tits,
*Groupes réductifs*, Inst. Hautes Études Sci. Publ. Math.**27**(1965), 55–150 (French). MR**207712**, DOI 10.1007/BF02684375 - Frederick P. Greenleaf, Martin Moskowitz, and Linda Preiss Rothschild,
*Unbounded conjugacy classes in Lie groups and location of central measures*, Acta Math.**132**(1974), 225–243. MR**425035**, DOI 10.1007/BF02392116 - Frederick P. Greenleaf, Martin Moskowitz, and Linda Preiss Rothschild,
*Compactness of certain homogeneous spaces of finite volume*, Amer. J. Math.**97**(1975), 248–259. MR**407196**, DOI 10.2307/2373670 - Siegfried Grosser and Martin Moskowitz,
*Compactness conditions in topological groups*, J. Reine Angew. Math.**246**(1971), 1–40. MR**284541**, DOI 10.1515/crll.1971.246.1
Harish-Chandra, - G. Hochschild,
*The structure of Lie groups*, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR**0207883** - G. D. Mostow,
*Homogeneous spaces with finite invariant measure*, Ann. of Math. (2)**75**(1962), 17–37. MR**145007**, DOI 10.2307/1970416 - David L. Ragozin and Linda Preiss Rothschild,
*Central measures on semisimple Lie groups have essentially compact support*, Proc. Amer. Math. Soc.**32**(1972), 585–589. MR**291373**, DOI 10.1090/S0002-9939-1972-0291373-4 - Kwan-Yuk Law Sit,
*On centralizers of generalized uniform subgroups of locally compact groups*, Trans. Amer. Math. Soc.**201**(1975), 133–146. MR**354923**, DOI 10.1090/S0002-9947-1975-0354923-2 - J. Tits,
*Automorphismes à déplacement borné des groupes de Lie*, Topology**3**(1964), no. suppl, suppl. 1, 97–107 (French). MR**158948**, DOI 10.1016/0040-9383(64)90007-2 - Tsuneo Tamagawa,
*On discrete subgroups of $p$-adic algebraic groups*, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 11–17. MR**0195864**

*Harmonic analysis on reductive $p$-adic groups*, Lecture Notes in Math., vol. 162, Springer-Verlag, Berlin and New York, 1970.

## Additional Information

- © Copyright 1975 American Mathematical Society
- 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