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CAT(-1)-spaces, divergence groups and their commensurators. (English) Zbl 0847.22004
The purpose of this paper is to study in detail the action of discrete groups on CAT(–1) spaces, a topological generalization of the class of manifolds of negative curvature. The main result is of the following nature, a topological analogue of Margulis’ superrigidity theorem: Let \(\Gamma\) be a discrete subgroup of a locally compact second countable group \(G\); let \(\text{Comm}_G (\Gamma)\) denote the commensurability group of \(\Gamma\) in \(G\). We suppose that \(\Gamma\) is such that there is a weak \((G, \Gamma' )\)-boundary for any \(\Gamma'\) of finite index in \(\Gamma\). Suppose \(\Lambda\) is a further group with \(G \subset \Lambda \subset \text{Comm}_G (G)\) so that \(\Lambda\) acts isometrically and ‘\(c\)-minimally’ on a CAT(–1) space \(Y\) (and the induced action of \(\Gamma\) on \(Y \) is non-elementary). Then the action of \(\Lambda\) on \(Y\) extends continuously to the closure of \(\Lambda\) in \(G\).
The authors consider in more detail the case where \(G\) is a product of Lie groups over local fields and obtain more precise results. Finally they investigate the case where \(G\) is the isometry group of a locally finite tree. This case arises when one considers the universal covering of a Cayley graph of a finitely presented group. This gives rise to some interesting examples, for example of a group \(\Gamma \subset \operatorname{Aut} (T)\) \((T\) a locally finite tree) where \(\text{Comm}_{\operatorname{Aut} (T)} (\Gamma)\) is dense in \(\operatorname{Aut} (T)\) and \(\operatorname{Aut} (T)\) is not discrete. The proofs involve studying in detail measure-theoretically the action of a group on a CAT(–1) space.

MSC:
22D40 Ergodic theory on groups
20E08 Groups acting on trees
22E40 Discrete subgroups of Lie groups
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