Abstract
PROBLEM 1.1. Show that the trivial representation of G is isolated in L2(F\G) in a quantitative way (independent of F). Let us point out right away that our problem is interesting for rank one groups only. For if G has real rank at least 2 then Kazhdan [12] has shown that the trivial representation of G is isolated in the unitary dual of G. This is called property T. Since representations occurring in L2(17\G) are unitary, we see the above problem is solvable for nonarithmetic reasons. If however G is Sp(n, 1) or the rank one real form of F4 then a result of Kostant [14] shows that the trivial representation is isolated in the unitary dual, again providing nonarithmetic answer to our Problem 1.1. The remaining rank one groups are essentially just SO,,1 and SU,,,l. If G = SL2(R) (which covers the identity component of SO2,1) Selberg's celebrated estimate
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