Please use this identifier to cite or link to this item:
Title: Identities on Hyperbolic Surfaces, Group Actions and the Markoff-Hurwitz Equations
Keywords: McShane's identity, Roger's dilogarithm, Mapping class group, Character variety, Coxeter group, Hurwitz equation
Issue Date: 19-Aug-2013
Citation: 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.
Appears in Collections:Ph.D Theses (Open)

Show full item record
Files in This Item:
File Description SizeFormatAccess SettingsVersion 
HengnanHuPhDThesis.pdf1.41 MBAdobe PDF



Page view(s)

checked on Dec 29, 2018


checked on Dec 29, 2018

Google ScholarTM


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.