The first part of this monograph is devoted to a
characterization of hypergeometric-like functions, that is,
"twists" of hypergeometric functions in "n"-variables. These are
treated as an ("n"+1) dimensional vector space of multivalued
locally holomorphic functions defined on the space of "n"+3 tuples
of distinct points on the projective line "P" modulo, the diagonal
section of Auto "P"="m." For "n"=1, the characterization may be
regarded as a generalization of Riemann's classical theorem
characterizing hypergeometric functions by their exponents at three
singular points.
This characterization permits the authors to compare monodromy
groups corresponding to different parameters and to prove
commensurability modulo inner automorphisms of "PU"(1, "n").
The book includes an investigation of elliptic and parabolic
monodromy groups, as well as hyperbolic monodromy groups. The
former play a role in the proof that a surprising number of
lattices in "PU"(1,2) constructed as the fundamental groups of
compact complex surfaces with constant holomorphic curvature are in
fact conjugate to projective monodromy groups of hypergeometric
functions. The characterization of hypergeometric-like functions by
their exponents at the divisors "at infinity" permits one to prove
generalizations in "n"-variables of the Kummer identities for "n"-1
involving quadratic and cubic changes of the variable.
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