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Math-Phys. Klasse, (1964), 145-164. [7] Wang, H. , 'On a Maximality Property of Discrete Subgroups with Fundamental Domain of Finite Measure, Amer. J. Math. 89 (1967), 124-132. [8] Yau, S. , 'Non-Existence of Continuous Convex Functions on Certain Riemannian Manifolds', Math. Ann. 207 (1974), 269-270. AVEZ HARMONIC FUNCTIONS ON GROUPS* ABSTRACT. Bounded harmonic functions on groups of non exponential growth are constant. INTRODUCTION E. Nelson [8] gave a remarkable proof of Liouville's theorem: Consider a bounded harmonic function I on Euclidean space.

And where the qi are integral polynomials in the lTw(ai, ... , b;). Now c', Pi and qi E ~(Sf), all integral, with the ql invariant by conjugation from G'. 7) says that c'(il')[M'] = (U : vft . E(il')[M'] is a K-characteristic number of M'. D. 53 FUNDAMENT AL DOMAINS 7. SUMMARY FOR FUNDAMENTAL DOMAINS We proved a number of integrality statements for volumes of fundamental domains, and here we assemble them for the case of simple groups. Remember, that the 2-sphere is hermitian and that the 4-sphere is the quaternion projective line.

E. t - s :s;; k. If we play the same game with z instead of y we get s:S;; k. 4 when n = 2. 6. Now let n be any integer; here we use [8, pp. 31-32]. 3 define matrices A(t) by A(t)(U2, ... , un) = (Y2(t), ... , Yn(t». 7 and [1, p. 254] we know that f(x, t) is nondecreasing. Following [8], the map t ~ A(t) is the solution of a differential equation A" + KA = 0; let us now fix some t :s;; c(g) and define B to be the solution of B" + KB = 0 with condition B(t) = 0 and B'(t) = - [ (the identity matrix).

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