Actions of discrete groups on complex projective spaces. (English) Zbl 1161.32301

Seade, José; Verjovsky, Alberto

Lyubich, M. (ed.) et al., Laminations and foliations in dynamics, geometry and topology. Proceedings of the conference held at SUNY at Stony Brook, USA, May 18–24, 1998. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-1985-2/pbk). Contemp. Math. 269, 155-178 (2001).

From the introduction: Holomorphic dynamics can be divided, roughly speaking, into three parts: iteration theory, actions of discrete groups

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and foliations, which includes actions of continuous Lie groups. Iteration theory and holomorphic foliations are being studied in all dimensions, while discrete groups actions

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have been essentially focused on Kleinian groups

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acting on the Riemann sphere

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, which is a fascinating area of mathematics. There are also very interesting results in higher dimensions, for example in [P. Deligne and D. G. Mostow, Commensurabilities among lattices in \(\text{PU}(1,n)\), Princeton, NJ: Princeton University Press (1993; Zbl 0826.22011)], about discrete subgroups

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of \(\text{U}(1,n)\) acting on the complex hyperbolic space

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. In [Math. Ann. 322, No. 2, 279–300 (2002; Zbl 1022.37032)] we introduced the concept of a complex Kleinian group; by this we mean a discrete subgroup of \(\text{PSL}(n+1,\mathbb C)\) acting on the projective space \(P_{\mathbb C}^n\) so that its region of discontinuity is non-empty, i.e., its limit set

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is not all of \(P_{\mathbb C}^n\). That seems to be a whole branch of mathematics waiting to be explored. This article is essentially an exposition of our article cited above; we have tried to present the main ideas of that article, removing most of the technical difficulties. Appropriate references are given in the test.
For the entire collection see [Zbl 0959.00033].

MSC:

32G05	Deformations of complex structures
20H10	Fuchsian groups and their generalizations (group-theoretic aspects)
32L25	Twistor theory, double fibrations (complex-analytic aspects)
32M05	Complex Lie groups, group actions on complex spaces
57N20	Topology of infinite-dimensional manifolds

Citations:

Zbl 0826.22011; Zbl 1022.37032