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By Werner Ballmann (auth.), Katsuhiro Shiohama, Takashi Sakai, Toshikazu Sunada (eds.)

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Extra info for Curvature and Topology of Riemannian Manifolds: Proceedings of the 17th International Taniguchi Symposium held in Katata, Japan, Aug. 26–31, 1985

Sample text

By §I, we have that there is a constant C > 0 such that ~l(Mi) > C only if there exists C > 0 such that h(Fi) > D, equivalent by §2 to the condition that, for some D' > 0, %1(L2(~i)) > D'. decompose L2(~ i) if and this last condition being into the orthogonal direct sum L2(~ i) = H l,j" We now It is standard from the representation theory of finite groups that each irreducible representation i of ~ occurs in the direct sum with a multiplicity equal to the dimension of the representation. Furthermore, the representational Laplacian also decomposes under this direct sum, so that the condition that ~l(L2(~i)) > D' %0(Hi,j) > D' is equivalent to i for all non-trivial irreducible representation Hi, j of ~ .

Representations and the discrete of PSL(2,Z/p) series fall into two classes: the (so named from analogy with the real case). e. that this action descends = ((-~ that X -IO)f)(x,y) to PSL(2,Z/p), = f(-x I - y) = ×(-i) we must have that f(x,y), is an even character. ) the f a' s to all Z/p x Z/p - {0} H X by the formula by (t). ,p-l, ~. We now can compute the on 30 (Vfa)( ~1 , a) = fa(~ , I - ~) 1 = fa(~ ' x~l) = x(l ~) 1 - _ fa(x_ll 1) , 1 1 (S)(fa)(X,t) = fa(l, -x) = X(-x)(fa(- ~ , i) = X(X)fa(- ~ , I) i or E(fa) = ~(a) f(- ~) = ×(x) f l(X,l).

X X k where X 0 is a Euclidean This decomposition ordering space and each X i is irreducible of de Rham is unique up to isometric for 1 < i < k . ,X k . Let ~ =MI x... x ~ be any Riemannian product d e c o m p o s i t i o n of a r simply connected manifold M of nonpositive sectional curvature, complete, and let F ~I(M) be any group of isometries. is said to be F-invariant {~i } are left invariant written uniquely if the foliations by F. In this case every element as ~ =¢I x... X ~ r projection homomorphisms 1

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