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Complex hyperbolic ideal triangle groups. (English) Zbl 0739.53055

Summary: Consider three distinct points \(u_ 1\), \(u_ 2\), \(u_ 3\) on the boundary of complex hyperbolic \(n\)-space and let \(C_ 1\), \(C_ 2\), \(C_ 3\) be the corresponding complex geodesics they span. Let \(\Gamma\) be the free product of three groups of order two and let \(\phi: \Gamma\to PU(n,1)\) be the homomorphism taking the generators to the inversions in \(C_ 1\), \(C_ 2\), \(C_ 3\). Conjugacy classes of such homomorphisms correspond to \(PU(n,1)\)-equivalence classes of triples (\(u_ 1\),\(u_ 2\),\(u_ 3\)). Such objects are parametrized by Cartan’s angular invariant \[ \mathbb{A}=\mathbb{A}(u_ 1,u_ 2,u_ 3)\in [-\pi/2,\pi/2]. \] For example \(\mathbb{A}=0\) if and only if \(u_ 1,u_ 2,u_ 3\) lie on the boundary of a totally real geodesic 2-plane; in this case \(\phi\) is a discrete embedding of \(\Gamma\) onto the subgroup of \(PO(2,1)\) generated by the reflections in the sides of an ideal triangle. At the other extreme, \(\mathbb{A}=\pm \pi/2\) if and only if \(u_ 1\), \(u_ 2\), \(u_ 3\) lie on the boundary of a single complex geodesic; in this case \(\phi(\Gamma)\) has order two. It is proved that if \(\phi\) is a discrete embedding, then \(| \tan\mathbb{A}| \leq \sqrt{125/3}\). Conversely, if \(| \tan\mathbb{A} |\leq \sqrt{35}\), then \(\phi\) is a discrete embedding. The proof of the latter involves the explicit construction of a Dirichlet fundamental polyhedron. The necessary condition \(| \tan\mathbb{A}|\leq \sqrt{125/3}\) is conjectured to be sufficient.

MSC:

53C56 Other complex differential geometry
53A35 Non-Euclidean differential geometry
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