  The axion-dilaton string effective action is expressed in Einstein frame
metric in a manifestly S-duality invariant form. It is shown that the moduli
can be redefined to describe surface of a 2+1 dimensional pseudosphere. The
classical cosmological solutions are of axion-dilaton are revisited. The
Wheeler De Witt equation for the system is exactly solved and complete set of
eigenfunctions are presented. The wave function factorizes and the one
depending on the moduli is obtained by appealing to the underlying S-duality
symmetry. Axion and dilaton parametrize the coset SU(1,1)/U(1), the S-duality
group, and the Hamiltonian is expressed as a sum of the Ricci scalar and the
Casimir of SU(1,1). Therefore, the complete set of wave functions, depending on
the moduli, is obtained from group theory technique. The evidence for the
existence of a W-infinity algebra in axion-dilaton cosmology is presented and
the origin of the algebra is primarily due to high degree of degeneracies in
the wave functions. It is qualitatively argued that axion-dilaton quantum
cosmology exhibits chaotic behaviour in the semiclassical limit.
