On uniform lattices in real semisimple groups

Bhagwat, Chandrasheel; Pisolkar, Supriya

doi:10.1090/proc/12961

On uniform lattices in real semisimple groups
HTML articles powered by AMS MathViewer

by Chandrasheel Bhagwat and Supriya Pisolkar PDF

Proc. Amer. Math. Soc. 144 (2016), 3151-3156 Request permission

Abstract:

In this article we prove that the co-compactness of the arithmetic lattices in a connected semisimple real Lie group is preserved if the lattices under consideration are representation equivalent. This is in the spirit of the question posed by Gopal Prasad and A. S. Rapinchuk in 2014 where instead of representation equivalence, the lattices under consideration are weakly commensurable Zariski dense subgroups.

References

Chandrasheel Bhagwat, Supriya Pisolkar, and C. S. Rajan, Commensurability and representation equivalent arithmetic lattices, Int. Math. Res. Not. IMRN 8 (2014), 2017–2036. MR 3194011, DOI 10.1093/imrn/rns282
A. Borel and J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math. 12 (1971), 95–104 (French). MR 294349, DOI 10.1007/BF01404653
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition. MR 0246136
H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank $1$ semisimple Lie groups, Ann. of Math. (2) 92 (1970), 279–326. MR 267041, DOI 10.2307/1970838
R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 235–252. MR 0249539
Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181, DOI 10.1007/BFb0079929
G. A. Margulis, On the arithmeticity of discrete groups, Dokl. Akad. Nauk SSSR 187 (1969), 518–520 (Russian); English transl., Soviet Math. Dokl. 10 (1969), 900–902. MR 0293007
M. Scott Osborne and Garth Warner, The theory of Eisenstein systems, Pure and Applied Mathematics, vol. 99, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 643242
Gopal Prasad and Andrei S. Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. Inst. Hautes Études Sci. 109 (2009), 113–184. MR 2511587, DOI 10.1007/s10240-009-0019-6
Gopal Prasad and Andrei S. Rapinchuk, Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces, Thin groups and superstrong approximation, Math. Sci. Res. Inst. Publ., vol. 61, Cambridge Univ. Press, Cambridge, 2014, pp. 211–252. MR 3220892
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
Nolan R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1992. MR 1170566