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.