  Neretin and Segal independently defined a semigroup of annuli with boundary
parametrizations, which is viewed as a complexification of the group of
diffeomorphisms of the circle. By extending the parametrizations to
quasisymmetries, we show that this semigroup is a quotient of the Teichmueller
space of doubly-connected Riemann surfaces by a Z action. Furthermore, the
semigroup can be given a complex structure in two distinct, natural ways. We
show that these two complex structures are equivalent, and furthermore that
multiplication is holomorphic. Finally, we show that the class of
quasiconformally-extendible conformal maps of the disk to itself is a complex
submanifold in which composition is holomorphic.
