# Algebraic Groups and Discontinuous Subgroups by A. Borel, G. Mostow PDF

By A. Borel, G. Mostow

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**Example text**

E. In the former hence x _< b I ~ a 5 and and again B(a,b) a p case, and and the a ~ b _< D ( a A b) = D ( p ) p _< a I v a 3. Similarly a i _< a and the (b I v x) since separation property relation. Since transitivity of E) B(a,b) holds. holds. or Otherwise, an a t o m atoms M(L) has the e x - [0,a I ~ a 2 ~ b I v P ] L either we h a v e All (since (b I v b3) A (a I ~ a 2 ~ p) # 0 a I ~ a 2 ~ p _< a in we have p _< b I v b3, b i _< b. aI ~ a2 ~ bI ~ b 2 = aI ~ bI ~ p fails, a6 < p _< a A b p _< b 2 v b 4 are under property); bI E b3 since A (a I ~ a 2 ~ p) _> p # 0, bI E x (b 3 v x) an atom E = A fails b2 E x (a I ~ a 2 ~ p) # 0.

A v = I, 1 E J(L), then a v = i v = i_ E ~(L). 2. If (ii) k Proof. 1, ducts of non-void (ii) ~ (i) ~ [0} U [i} But (iv) follows The is a complete lattice. Moreover ~(L) U [0} U [i} if these and sums in L. from Lemma to show that ~(L) U [0} is closed under pro- ~ ~ S=J(L) U [0}. U[O}, [0}. in (1) c Immediate subsets. ScJ(L) (iii). is a complete Suppose a £ J ( L ) U ~O} U ~i}. a ~J(L)U of sets for are equivalent: (iii) J ( L ) U [O} U ~i} then ~products (ili) it suffices By hypothesis, course is maximal; are the same as in Lemma S~.

U[O}, [0}. in (1) c Immediate subsets. ScJ(L) (iii). is a complete Suppose a £ J ( L ) U ~O} U ~i}. a ~J(L)U of sets for are equivalent: (iii) J ( L ) U [O} U ~i} then ~products (ili) it suffices By hypothesis, course is maximal; are the same as in Lemma S~. L E ~r' then the following are satisfied ~(L) U [0} U [l} rings are maximal. 1. then U~O}. (iii) ~ Let S = ~IJ If from Lemma a=NS. 2. 59 6. THE CASE THAT J(L) We note that for complete lattice AND ~(L) LE2r ARE COMPLETE LATTICES FOR the condition that algebra.

### Algebraic Groups and Discontinuous Subgroups by A. Borel, G. Mostow

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