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

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

2011-11-16

arxiv.org/abs/1111.3963v2

Physics

Mathematical Physics

20 pages, some cosmetic changes

Scientific paper

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ 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 $n_o$. We propose the dual graph description of the corresponding Teichm\"uller spaces, construct the Poisson algebra of the Teichm\"uller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations.

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