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On discontinuous action of monodromy groups on the complex n-ball. (English) Zbl 0657.22014
There is defined for each \((n+3)\)-tuple \((\mu_ 1,...,\mu_{n+3})\) of nonintegral real numbers with \(\sum \mu_ i\) integral, a subgroup \(\Gamma_{\mu}\) acting on a complex \((n+1)\)-dimensional vector space \(V_{\mu}\), which preserves a hermitian form \(<, >\). The author is concerned principally with the case that the signature of this form is (l,n) (l positive, n negative). In this case the image of \(V^+=\{v\in V_{\mu}\), \(<v,v>>0\}\) in the projective space \(P^ n\) associated to \(V_{\mu}\) is the complex ball \(B^ n\). The subgroup \(\Gamma_{\mu}\) associated to the \((n+3)\)-tuple \((\mu_ 1,...,\mu_{n+3})\) is generated by complex reflections.
There is a classical problem: when the subgroup \(\Gamma_{\mu}\) will be a lattice in PU(1,n), and so one can deal with the action of \(\Gamma_{\mu}\) on \(B^ n\) and \(P^ n\), e.d. \(\Gamma_{\mu}\subset PU(1,n)\). If \(\mu\) is a disc \((n+d)\)-tuple satisfying condition INT: for all \(i\neq j\) such that \(\mu_ i+\mu_ j<1\), \((1-\mu_ i-\mu_ j)^{- 1}\in {\mathbb{Z}}\), then \(\Gamma_{\mu}\) is a lattice of PU(1,n). Subsequently the hypothesis INT was weakened to condition \(\Sigma\) INT: there is a subset \(S_ 1\subset \{1,2,...,n+3\}\) such that \(\mu_ i=\mu_ j\) for all \(i,j\in S_ 1\) and moreover, for all \(i\neq j\) such that \(\mu_ i+\mu_ j<1\) \((1-\mu_ i-\mu_ j)^{-1}\in (1/2){\mathbb{Z}}\), if \(i,j\in S_ 1\) and \(\in {\mathbb{Z}}\) otherwise. One of the main results of this paper asserts that if \(\Gamma_{\mu}\) is discrete in Aut \(B^ n\), \(n>3\), then \(\mu\) satisfies condition \(\Sigma\) INT.
Reviewer: G.A.Soifer

MSC:
22E40 Discrete subgroups of Lie groups
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
20H15 Other geometric groups, including crystallographic groups
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