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arXiv · 1001.5211

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

David Radnell·Eric Schippers
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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.

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David Radnell, Eric Schippers. 2010-01-28. The semigroup of rigged annuli and the Teichmueller space of the annulus. https://doi.org/10.1112/jlms%2Fjds006

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