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

TL;DR

This paper explores the complex structure of a semigroup of annuli with boundary parametrizations, linking it to the Teichmueller space of doubly-connected Riemann surfaces, and establishes holomorphic properties of related maps.

Contribution

It demonstrates that the semigroup is a quotient of the Teichmueller space by a Z action and shows the equivalence of two natural complex structures on this semigroup.

Findings

The semigroup is a quotient of the Teichmueller space of doubly-connected surfaces.

Two natural complex structures on the semigroup are equivalent.

Multiplication in the semigroup is holomorphic.

Abstract

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.