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Traces in complex hyperbolic geometry

Parker, John R.

Traces in complex hyperbolic geometry Thumbnail



William M. Goldman

Caroline Series

Ser Peow Tan


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.


Parker, J. R. (2012). Traces in complex hyperbolic geometry. In W. M. Goldman, C. Series, & S. P. Tan (Eds.), Geometry, topology and dynamics of character varieties (191-245).

Conference Name Geometry, Topology and Dynamics of Character Varieties
Conference Location National University of Singapore
Publication Date Aug 1, 2012
Deposit Date Jun 25, 2012
Publicly Available Date Feb 21, 2014
Publisher World Scientific Publishing
Pages 191-245
Series Title Lecture notes series, Institute for Mathematical Sciences, National University of Singapore
Series Number 23
Book Title Geometry, topology and dynamics of character varieties.
Keywords Complex hyperbolic space, Trace, Invariants.
Public URL


Accepted Conference Proceeding (442 Kb)

Copyright 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.

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