  This monograph is an updated and extended version of the author's PhD thesis.
It consists of an introductory text followed by two separate parts which are
loosely related but may be read independently of each other. In Part I we
analyze certain hyperbolic structures arising when studying gravity in the
vicinity of a spacelike singularity (the "BKL-limit"). In this limit, spatial
points decouple and the dynamics exhibits ultralocal behaviour which may be
described in terms of a (possibly chaotic) hyperbolic billiard. In all
supergravities arising as low-energy limits of string theory or M-theory, the
billiard dynamics takes place within the fundamental Weyl chambers of certain
hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden
infinite-dimensional symmetries of the theory. Part II of the thesis is devoted
to a study of how (U-)dualities in string theory provide powerful constraints
on perturbative and non-perturbative quantum corrections. These dualities are
described by certain arithmetic groups G(Z) which are conjectured to be
preserved in the effective action. The exact couplings are given by automorphic
forms on the double quotient G(Z)\G/K. We discuss in detail various methods of
constructing automorphic forms, with particular emphasis on non-holomorphic
Eisenstein series. We provide detailed examples for the physically relevant
cases of SL(2,Z) and SL(3,Z), for which we construct their respective
Eisenstein series and compute their (non-abelian) Fourier expansions. We also
show how these techniques can be applied to hypermultiplet moduli spaces in
type II Calabi-Yau compactifications, and we provide a detailed analysis for
the universal hypermultiplet.
