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Maximal convergence groups and rank one symmetric spaces. (English) Zbl 1162.30028

The authors consider rank one symmetric spaces \(\mathcal H\) of non-compact type, excluding the hyperbolic plane. That is, they consider complex hyperbolic spaces of complex dimension one. The boundary \(\partial \mathcal H\) of \(\mathcal H\) is canonically endowed with a conformal structure, and there is an isomorphism between the group of conformal homeomorphisms Conf\((\partial \mathcal H)\) and the group Isom\((\mathcal H)\) of all isometries of \(\mathcal H\). The main result of the paper gives a generalisation of a result by [F. W. Gehring and G. J. Martin, Proc. Lond. Math. Soc., III. Ser. 55, 331–358 (1987; Zbl 0628.30027)]. More precisely, the main result of the paper is: \[ \text{ Conf\((\partial \mathcal H)\) is a maximal convergence group.} \] Here, convergence group means that for each infinite sequence \((g_{n})\) in Conf\((\partial \mathcal H)\) there exists a subsequence which converges uniformly to an element of Conf\((\partial \mathcal H)\) such that there exist \(z,w \in \partial \mathcal H\) (\(z\) the attractor, and \(w\) the repeller) for which \((g_{n})\) converges uniformly to the constant function \(z\), uniformly outside each neighbourhood of \(w\). Moreover, maximal refers to that if \(G \supseteq\) Conf\((\partial \mathcal H)\) is a convergence group, then \(G=\) Conf\((\partial \mathcal H)\).

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)

Citations:

Zbl 0628.30027
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References:

[1] Gromov, Lecture Notes in Math. pp 39–
[2] Gehring, Proc. London Math. Soc. 55 pp 331– (1987) · Zbl 0628.30027
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