Teichmuller Space of a Countable Set of Points on the Riemann Sphere

Abstract

We introduce the Teichm\"uller space $T(E)$ of an ordered countable set $E$

of infinite number of distinct points on the Riemann sphere.

We discuss the relation between the Teichm\"uller distance on $T(E)$

and a natural one on the configuration space for $E$.

Also we give a system of global holomorphic coordinates

for $T(E)$ when $E$ is determined from a finitely generated semigroup

consisting of M\"obius transformations

with the totally disconnected forward limit set.