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Traces in complex hyperbolic geometry.

Parker, John R. (2012) 'Traces in complex hyperbolic geometry.', in Geometry, topology and dynamics of character varieties. Singapore: World Scientific, pp. 191-245. Lecture notes series, Institute for Mathematical Sciences, National University of Singapore. (23).


We discuss the relationship between the geometry of complex hyperbolic manifolds and orbifolds and the traces of elements of the corresponding subgroup of SU(2, 1). We begin by showing how geometrical information about individual isometries is encoded by their trace. We then consider traces for groups Γ of isometries in two specific cases. First, we consider the case where Γ is a free group on two generators, which we view as the fundamental group of a three holed sphere. We indicate how to use this analysis to give complex hyperbolic Fenchel-Nielsen coordinates. Secondly, we consider the case where Γ is a triangle group generated by complex reflections in three complex lines. We keep in mind similar results from the more familiar setting of Fuchsian and Kleinian groups and we explain those examples from our point of view.

Item Type:Book chapter
Additional Information:This paper was presented at the Summer School on Geometry, Topology and Dynamics of Character Varieties held at the National University of Singapore's Institute for Mathematical Sciences in July 2010.
Keywords:Complex hyperbolic space, Trace, Invariants.
Full text:(AM) Accepted Manuscript
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Publisher statement:Parker, John R. (2012) 'Traces in complex hyperbolic geometry.', in Geometry, topology and dynamics of character varieties, edited by William Goldman (University of Maryland, USA), Caroline Series (University of Warwick, UK), Ser Peow Tan. Copyright © 2012 with permission from World Scientific Publishing Co. Pte. Ltd.
Date accepted:No date available
Date deposited:21 February 2014
Date of first online publication:August 2012
Date first made open access:No date available

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