Please use this identifier to cite or link to this item: http://scholarbank.nus.edu.sg/handle/10635/48648
Title: Identities on Hyperbolic Surfaces, Group Actions and the Markoff-Hurwitz Equations
Authors: HU HENGNAN
Keywords: McShane's identity, Roger's dilogarithm, Mapping class group, Character variety, Coxeter group, Hurwitz equation
Issue Date: 19-Aug-2013
Source: HU HENGNAN (2013-08-19). Identities on Hyperbolic Surfaces, Group Actions and the Markoff-Hurwitz Equations. ScholarBank@NUS Repository.
Abstract: This thesis is mainly focused on identities motivated by McShane's identity. Firstly, by applying the Luo-Tan identity, we derive a new identity for a hyperbolic one-holed torus T. Secondly, we review the $SL(2,\mathbb{C})$ character variety X of $\pi_1(T) = < a, b > $ and the action of the mapping class group MCG of T on X. The Bowditch Q-conditions describe an open subset of X on which MCG acts properly discontinuously. We prove a simple and new identity for characters satisfying the Bowditch Q-conditions which generalizes McShane's identity. Thirdly, we can interpret the action of MCG on X as the action of the Coxeter group $G_3$ on \mathbb{C}^3 leaving invariant the varieties $x_1^2 + x_2^2 + x_3^2 = x_1 x_2 x_3 + \mu$, where $\mu \in \mathbb{C}$. We generalize the study to the action of the Coxeter group $G_m$ on $\mathbb{C}^m$, where $m \ge 4$, which leaves invariant the varieties described by the Hurwitz equations $x_1^2 + x_2^2 +? + x_m^2 = x_1 x_2? x_m + \mu$. We formulate a generalization of the Bowditch Q-conditions and show that it describes an open subset of $\mathbb{C}^m$ on which $G_m$ acts properly discontinuously. Finally, we prove an identity for the orbit of any m-tuple in the subset.
URI: http://scholarbank.nus.edu.sg/handle/10635/48648
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