Traces in complex hyperbolic geometry

Parker, John R.

Abstract

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.

Citation

Parker, J. R. (2012, August). Traces in complex hyperbolic geometry. Presented at Geometry, Topology and Dynamics of Character Varieties, National University of Singapore

• Presentation Conference Type: Conference Paper (published)
• Conference Name: Geometry, Topology and Dynamics of Character Varieties
• 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.
• DOI: https://doi.org/10.1142/9789814401364_0006
• Keywords: Complex hyperbolic space, Trace, Invariants.
• Public URL: https://durham-repository.worktribe.com/output/1157570