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Isomorphisms among monodromy groups and applications to lattices in \(PU(1,2)\). (English) Zbl 0759.22013

This paper completes previous work of G. D. Mostow [Pac. J. Math. 86, 171-276 (1980; Zbl 0456.22012); Publ. Math., Inst. Hautes Etud. Sci. 63, 91-106 (1986; Zbl 0615.22009); Bull. Am. Math. Soc. 16, 225-246 (1987; Zbl 0639.22005); On discontinuous action of monodromy groups on the complex \(n\)-ball (to appear)] and P. Deligne and G. D. Mostow [Publ. Math., Inst. Hautes Etud. Sci. 63, 5-89 (1986; Zbl 0615.22008)]. In these previous works, certain monodromy subgroups of \(PU(1,n)\), related to hypergeometric differential systems, were defined and their possible discreteness was investigated. These groups arise as monodromy groups of the Picard-Fuchs equations obtained when the \(n + 3\) ramification points of a covering of \(\mathbb{P}^ 1\), with fixed local monodromy, are moved around, precisely as in the classical theory of ordinary hypergeometric functions. Mostow gave a necessary and sufficient condition for these groups to be discrete, holding when \(n>3\). This paper deals with the case \(n=2\). After a clear review of previous work, the author, inspired by some computer-work, completes the list of discrete groups arising for \(n=2\). He can then verify that the Mostow condition essentially holds also in the present case. He then computes the volumes of the fundamental domains for all of his discrete groups, and subsequently determines relative indexes, in case of mutual inclusions. He also proves certain isomorphisms among various groups. His conclusions are nicely summarized in several tables that include the results of Mostow and Deligne. The paper contains very explicit proofs of its statements, and should be of great help to both specialists and newcomers of the topic.

MSC:

22E40 Discrete subgroups of Lie groups
11F06 Structure of modular groups and generalizations; arithmetic groups
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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