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Isometry groups of proper CAT(0)-spaces of rank one. (English) Zbl 1275.20047
Let \(X\) be a proper CAT(0) metric space. An isometry \(g\in\mathrm{Iso}(X)\) is called rank-one if it has an axis which does not bound a flat half-plane. For example, if \(X\) is a hyperbolic space then any axial isometry of \(X\) is rank-one. The general notion of rank-one isometries goes back to an important paper by W. Ballmann and M. Brin [Publ. Math., Inst. Hautes Étud. Sci. 82, 169-209 (1995; Zbl 0866.53029)].
The author shows that if \(G\) is a non-elementary closed subgroup of \(\mathrm{Iso}(X)\) that contains a rank-one element then, up to passing to an open subgroup of finite index, either \(G\) is a compact extension of a totally disconnected group or \(G\) is a compact extension of a simple Lie group of rank one. This theorem follows from a non-vanishing theorem for the second continuous bounded cohomology of \(G\) that extends a prior result of the author for isometry groups of proper hyperbolic metric spaces [Geom. Funct. Anal. 19, No. 1, 170-205 (2009; Zbl 1273.53037)]. The non-vanishing theorem also gives a superrigidity theorem for representations of irreducible lattices in higher rank semi-simple Lie groups into \(\mathrm{Iso}(X)\) as a corollary. These results are closely related to the previous work of N. Monod, Y. Shalom, I. Mineyev, M. Bestvina, K. Fujiwara, and P.-E. Caprace (see the bibliography and discussion in the paper).

20F67 Hyperbolic groups and nonpositively curved groups
53C24 Rigidity results
22E46 Semisimple Lie groups and their representations
22E41 Continuous cohomology of Lie groups
53C35 Differential geometry of symmetric spaces
20J06 Cohomology of groups
Full Text: DOI
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