Traces in Complex Hyperbolic Triangle Groups

Details Traces in Complex Hyperbolic Triangle Groups Traces in Complex Hyperbolic Triangle Groups

2004-02-10

arxiv.org/abs/math/0402153v3

Geom. Dedicata 111 (2005), 159-185.

Mathematics

Differential Geometry

22 pages, 1 figure; prop. 11 added, typos corrected; cor. 19 removed (not correct)

Scientific paper

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 alpha 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 exp(i alpha) with coefficients independent of alpha 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 mu-reflections. We apply these formulas to prove some discreteness and some non-discreteness results for complex hyperbolic triangle groups.

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