Existence of nonarithmetic monodromy groups.
In an 1885 paper, E. Picard defined a subgroup Tau(Lambda) of PU(2,1) generated by monodromies and depending on parameters Lambda = (lambda(1),lambda(2),lambda(3),lambda(4)), 0 < lambda(i) < 1, < lambda(i) < 3, lambda(i) + lambda(j) >/= 1, 1 </= i < j </= 4. The family Tau(Lambda) resembles the family of groups Tau([unk]) defined in 1978 but is a different family. In common with the groups Tau([unk]), (i) Tau(Lambda) is discrete for a finite number of Lambda, (ii) Tau(Lambda) is a nonarithmetic lattice for some Lambda, and (iii) for all Lambda in [unk](4), there is a compact complex surface S(Lambda) with pi(1) [S(Lambda)] of finite index in Tau(Lambda).
Mostow
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Institut des Hautes Etudes Scientifiques, Bures-Sur-Yvette, France, 91440.