The semigroup of rigged annuli and the Teichmüller space of the annulus

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 Teichmüller 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.