By Edwin Weiss

ISBN-10: 0486401898

ISBN-13: 9780486401898

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**Sample text**

8, tr X(R) also . Whenever denotes element depends R half R° on the is an ar- the o r d e r (depend- corresponding F-conjugacy to class only. Proof. As are not elliptic ~lell(Z) x(M) j M = x ( - M ) j _ M = fixed Z points Z (M £ F), we of have for all z 6 IH which F : tr X(S-IRS) JS_IRs (z) H(z, S-1RSz) ks(~(z, S-IRSz)) • {R} r S6Z (Rl\r 0<8<. The inner sum runs through right cosets ment R 6 {R} F . 10) X(S-IRs) J and a complete system centralizer Z (R) in the p r o o f of Prop. 5, of the selected it f o l l o w s I (z) H(z,S-IRSz) = x(S) -I x(R) x(S) JR(SZ) H(Sz,RSz), S-RS from eleDef.

6 A~IUmKI proved (by c h o i c e of < g . Im A I Sz Im A 2 S z unique norm . As z' 6 a x ( P ) , N AIIumK I . the Im A I S Z least < Im AISZ' x ]y1,Y1+1] 6 ax(Q and . of largest and is m e t at the sub- S Yl +I has be is a c o m p a c t z 6 IH} the 6 S ax(P) ax(P) m I} trace points £ ~ a x ( A I Q A ~ I) = A I S Let on z 6 ax(P) c K I . The Yl + . := SPS -I Im A I S Z positive lies ' m Suppose s ~ Im A I S Z AISZ z the {P]F ¢ ~ < yj = inf{Iz,Pzl: iff both 0 AjIumKj the case. is v a l i d exist with logN(P) joining z × ]yj,yj+l ]~/ > O , j = I .....

Finite has an e x p a n s i o n absolutely of is a l w a y s set. 3 w e r e the form cofinite Sarnak ([DIPS],[PSI],[PS2]) of the f = E (fn,f) "fn nZO the in recent work (cf. which to be e i t h e r question will This would question in- is still at the be a t r i v i a l Iwaniec, both exist the h y p e r b o l i c be d i s c u s s e d of D e s h o u i l l e r s , that a countably if t h e r e of [Ro 3]). conjecture suggests ]H. end of conse- Phillips conjectures and might well false. 1 multiplier is an u n b o u n d e d > I tribution D group Roelcke's of it.

### Algebraic Number Theory by Edwin Weiss

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