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SIGMA 3 (2007), 066, 37 pages math.AG/0610872
https://doi.org/10.3842/SIGMA.2007.066
Contribution to the Vadim Kuznetsov Memorial Issue
Teichmüller Theory of Bordered Surfaces
Leonid O. Chekhov
Steklov Mathematical Institute, Moscow, Russia
Institute for Theoretical and Experimental Physics, Moscow, Russia
Poncelet Laboratoire International Franco-Russe, Moscow, Russia
Concordia University, Montréal, Quebec, Canada
Received January 05, 2007, in final form April 28, 2007; Published online May 15, 2007
Abstract
We propose the graph description of Teichmüller theory of surfaces
with marked points on boundary components (bordered surfaces).
Introducing new parameters, we formulate this theory in terms of
hyperbolic geometry. We can then describe both classical and quantum
theories having the proper number of Thurston variables
(foliation-shear coordinates), mapping-class group invariance (both
classical and quantum), Poisson and quantum algebra of geodesic
functions, and classical and quantum braid-group relations. These
new algebras can be defined on the double of the corresponding graph
related (in a novel way) to a double of the Riemann surface (which
is a Riemann surface with holes, not a smooth Riemann surface). We
enlarge the mapping class group allowing transformations relating
different Teichmüller spaces of bordered surfaces of the same
genus, same number of boundary components, and same total number of
marked points but with arbitrary distributions of marked points
among the boundary components. We describe the classical and quantum
algebras and braid group relations for particular sets of geodesic
functions corresponding to An and Dn algebras and discuss
briefly the relation to the Thurston theory.
Key words:
graph description of Teichmüller spaces; hyperbolic geometry; algebra of geodesic functions.
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