By Xiangdong Xie

We identify a Bowen kind tension theorem for the elemental crew of a noncompacthyperbolic manifold of finite quantity (with measurement at the least 3).

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1. Again this contradicts (vii) The final contradiction. By the above results X = EU, E CE(t) is dihedral of order q - E, ~ 1 (q), q odd. Therefore Z where q '" (mod 4), E = ±1, T is of type (2, 2). If U;: E we are done because E = O*(X), CU(T) = U n CE(T) = T nu = 1 and U = O(CE(t)) is cyclic. As U = CU(T)[U, T] and [U, T];: E, it follows that CU(T)"* 1. Let u E CU(T) have prime order p. Let q = rn, where r is a prime. First u must induce a field automorphism on E, since CE(T) = T. CE(u) ~ Thus m L (r ) where m = nip.

If then t were to induce the 'transpose inverse' automorphism on L, a Sylow 2-subgroup of L(t) would be non-abelian. Thus t must induce an inner automorphism T on L if L ~ L (q). z Since CL (T) is dihedral if L ~ L (q), q odd, and an elementary z n abelian 2-group if L ~ L (2 ), CL (T) normalizes no non-solvable sub- z group of L except L itself. Here T is an involution in L. Hence K = L. If L is of type JR, then t centralizes a Sylow 2-subgroup T n L of L and so t normalizes CL (T) for T E T n L.

1 1 n. The groups E. l 1 is a non-abelian simple group. 1 They are called the components of X. We will frequently write E. ). 1 34 < i< - 1 1 1 Lemma 10. 2. ]:::: 1 if i J 1 oF j. (b) E(X):::: El ... En' Proof. Z(E(X))/Z(E(X)). ] :::: 1 if i J 1 oF Thus j. ]:::: 1. 1 J 1 Since E. ] :::: 1. 1 J 1 (b) Clearly E(X):::: E ... E Z(E(X)). 1 n Hence E(X):::: E(X)' :::: (E ... E )' :::: E ... E. 1 n 1 n the proof. 1/ This completes The most remarkable property of the groups E. is contained in 1 the next Lemma.