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Arithmetics Groups by Humphreys J.E.

By Humphreys J.E.

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Id) : Rk ;! R g Rk where i is the embedding of the closed submanifold Graph into R ( id) C 1 (R Since 0 = r(0) = Rk f0g) = g i C 1 (R Rk Rk . Therefore f0g): (0) = (0), we see that g(0) = 0. So we also have C 1 (Rk f0g) = g C 1 (Graph f0g): Therefore it remains to prove that i C 1 (R f0g) = C 1 (Graph f0g): Now take an arbitrary f 2 C 1 (Graph f0g). There is a smooth extension f~ of f Rk on R Rk but it need not be at at zero. So consider a submanifold chart ( U ) of Graph around 0 and de ne fU : U ;!

We de ne the normal slice at x by ; Sx := expx Norr=2(G:x) x : July 31, 1997 P. 2 6. 3. Lemma. Under these conditions we have (1) (2) Sg:x = g:Sx . Sx is a slice at x. Proof. 1(1) : ; ; ; Sg:x = expg:x Tx`g Norr=2 (G:x) x = `g expx Norr=2 (G:x) x = g:Sx (2) r : G:Sx ;! G:x : expg:x X 7! g:x de nes a smooth equivariant retraction (note that Sx and Sy are disjoint if x 6= y). 4. De nition. Let M be a G-manifold and x 2 M , then there is a representation of the isotropy group Gx Gx ;! GL(TxM ) : g 7!

Again, it will do to nd a slice S at x with only a nite number of Gx -orbit types. If dim S < dim M , this follows from the induction hypothesis. Now suppose dim S = n. S is equivariantly di eomorphic to an open ball in Tx M under the slice representation (by exp). Since the slice representation is orthogonal, it restricts to a Gx -action on the sphere S n;1 . By the induction hypothesis, locally, S n;1 has only nitely many Gx -orbit types. Since S n;1 is compact, it has only nitely many orbit types globally.

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