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Math. Franee 94 (1966), 97-140. M. Mars, The Tamagawa number of 557-574. 2An , Ann. Math. 89 (1969), [30] D. Meuser, On the rationa1ity of eertain generating funetions, Math. Ann. 256 (1981), 303~310. [31] D. Meuser, On the poles of a loea1 zeta_funetion for eurves, Invent. math. 73 (1963), 445-465. D. Mostow, Self-adjoint groups, Ann. Math. 62 (1955), 44-55. [33] T. Ono, On the relative theory of Tamagawa numbers, Ann. Math. 82 (1965), 88-111. [34] T. Ono, An integral attaehed to a hypersurfaee, Amer.

Would then have established that C(M), P(M) H(M x R) and We are singu- lar for their natural local analytic structures. To obtain the above results concerning the spaces of structures it is convenient to replace them with the space of classes of representations of r, the fundamental group of groups of the model space. M, into the automorphism This is possible because of the following general result. Let S(M) be aspace of marked locally homogeneous structures X = G/H. 11 of Lok [13] states that G acts by conjugation.

P(M). Our main Also first author was partially supported by NSF grant #MCS77-24l03, the second by NSF grant #MCS-8200639. 49 H(M x R) is an interesting space closely related to C(M). Dur first main result is a lower bound for the dimensions of the three previous deformation spaces by r, the largest number of disjoint, non-singular, totally geodesic hypersurfaces contained in surface of genus g then r = 3g - 3. M. If M is a hyperbolic From this bound, it is easily shown that the deformation spaces have arbitrarily large dimension as M varies.

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