Traces in complex hyperbolic triangle groups. (English) Zbl 1115.32015
Summary: 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 \(e^{i\alpha}\) 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.
MSC:
| 32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
| 51F15 | Reflection groups, reflection geometries |
| 53C35 | Differential geometry of symmetric spaces |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
Keywords:
complex hyperbolic geometry; trace formulas; complex hyperbolic triangle groups; complex reflections; triangles in the complex hyperbolic plane; winding numbersReferences:
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| [14] | Schwartz, R. E.: Complex hyperbolic triangle groups, In: Proceedings of the International Congress of Mathematicians, Vol. II, Beijing, 2002, pp. 339–349. · Zbl 1022.53034 |
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| [17] | Wyss-Gallifent, J.: Complex Hyperbolic Triangle Groups, Ph.D. thesis, University of Maryland, (2000). |
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