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# A 1.5-dimensional version of Hopfs Theorem on the number of by Bieri R. By Bieri R.

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8. #2. Show that if G is semisimple, and N is any closed, normal subgroup of G, then G/N is semisimple. 3D. THE SIMPLE LIE GROUPS #3. 31 a) Show that if φ : G → A is a continuous homomorphism, and A is abelian, then φ is trivial. b) Show that [G, G] = G. #4. Show that a connected Lie group H is semisimple if and only if H has no nontrivial, connected, solvable, normal subgroups. ] #5. Suppose N is a connected, closed, normal subgroup of G = G1 × · · · × Gr , where each Gi is simple. Show that there is a subset S of {1, .

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20 CHAPTER 2. 3) Theorem. R-rank(G) is the largest natural number r , such that X contains a closed, simply connected, r -dimensional flat. For example, if X is compact, then every closed, totally geodesic, flat subspace of X must be a torus, not Rn , so R-rank(G) = 0. On the other hand, if X is not compact, then X has unbounded geodesics (for example, if X is irreducible, then every geodesic goes to infinity), so R-rank(G) ≥ 1. Therefore: R-rank(G) = 0 X is compact. Thus, there is a huge difference between R-rank(G) = 0 and R-rank(G) > 0, because no one would mistake a compact space for a noncompact one.