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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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