Geometriae Dedicata

, Volume 111, Issue 1, pp 159–185

Traces in Complex Hyperbolic Triangle Groups


DOI: 10.1007/s10711-004-1493-0

Cite this article as:
Pratoussevitch, A. Geom Dedicata (2005) 111: 159. doi:10.1007/s10711-004-1493-0


We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant α of triangles in the complex hyperbolic plane. The main result of the paper is a formula, which expresses the trace of an element of the group as a Laurent polynomial in ei α with coefficients independent of α and computable using a certain combinatorial winding number. We also give a recursion formula for these Laurent polynomials and generalise the trace formulas for the groups generated by complex μ-reflections. We apply these formulas to prove some discreteness and some non-discreteness results for complex hyperbolic triangle groups.

Mathematics Subject Classification (2000)

Primary 51M10Secondary 32M1553C5553C35


complex hyperbolic geometrytriangle groups

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany