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 Int. Math. Res. Not. IMRN, 2014, Volume 2014, Issue 10, Pages 2746–2772 (Mi imrn7)

Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables

L. Chekhovab, M. Shapiroc

a Laboratoire Poncelet and Steklov Mathematical Institute, Moscow, Russia
b School of Mathematics, Loughborough University, Loughborough, Leicestershire LE11 3T UK
c Department of Mathematics, Michigan State University, East Lansing, MI, USA

Abstract: We define a new generalized class of cluster-type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos\pi/n_0+x^{-1}$ these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders $n_0$. In the second part of the paper, we propose the dual graph description of the corresponding Teichmüller spaces, construct the Poisson algebra of the Teichmüller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations thus providing the complete description of the above Teichmüller spaces.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-0044011-01-12037 Ministry of Education and Science of the Russian Federation NSh-4612.2012.1 National Science Foundation DMS-0800671DMS-1101369 This work was supported in part by the Russian Foundation for Basic Research (Grant Nos. 11-01-00440-a and 11-01-12037-ofi-m-2011), by the Grant of Supporting Leading Scientific Schools of the Russian Federation NSh-4612.2012.1, and by the Program Mathematical Methods for Nonlinear Dynamics (to L.Ch.) and was supported in part by grants DMS-0800671 and DMS-1101369 (to M.S.).

DOI: https://doi.org/10.1093/imrn/rnt016

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Document Type: Article
Accepted:09.01.2013
Language: English