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

Abstract

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 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.

20 pages, notations and many essential typos corrected, most significantly, formulae 2.3, 2.5, proof of Lemmata 2.6 and 4.5. Journal reference is added (published version contains typos)

References in corpus (3)

• Cluster algebras and triangulated orbifolds
• Closed/open string diagrammatics
• Teichmüller Theory of Bordered Surfaces

Cited by in corpus (17)

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• Filling of closed Surfaces
• Rogers dilogarithms of higher degree and generalized cluster algebras
• Symplectic structures on Teichmüller spaces ${\mathfrak{T}}_{g,s,n}$ and cluster algebras
• Quantum generalized cluster algebras and quantum dilogarithms of higher degrees
• Scattering diagrams for generalized cluster algebras
• Multiple generalized cluster structures on $D({\text{GL}}_n)$
• Generalized Cluster Structures Related to the Drinfeld Double of $G{L}_n$
• Snake Graphs and Caldero-Chapoton Functions from Triangulated Orbifolds
• A Generalization of Markov Numbers
• Orbifold diagrams
• Starfish lemma via birational quasi-isomorphisms
• Cluster Monomials in Graph Laurent Phenomenon Algebras