It is well-known that a point T 2 cv N in the (unprojectivized) Culler-Vogtmann Outer space cv N is uniquely determined by its translation length function k k T W F N ! R. A subset S of a free group F N is called spectrally rigid if, whenever T; T 0 2 cv N are such that kgk T D kgk T 0 for every g 2 S then T D T 0 in cv N . By contrast to the similar questions for the Teichmüller space, it is known that for N 2 there does not exist a finite spectrally rigid subset of F N .
All σ-compact, locally compact groups acting sharply n-transitively and continuously on compact spaces M have been classified, except for n = 2, 3 when M is infinite and disconnected. We show that no such actions exist for n = 2 and that these actions for n = 3 coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary. We further give a characterization of non-compact groups acting 3-properly and transitively on infinite compact sets as non-elementary boundary transitive hyperbolic groups. The main tool is a generalization to locally compact groups of Bowditch's topological characterization of hyperbolic groups. Finally, in contrast to the case n = 3, we show that for n ≥ 4 if a locally compact group acts continuously, n-properly and n-cocompactly on a locally connected metrizable compactum M , then M has a local cut point.
Abstract. In this article, we study the outer automorphism group of a group G decomposed as a finite graph of groups with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by taking extensions of relative automorphism groups of vertex groups, groups of Dehn twists and groups of automorphisms of free products. We apply this description and obtain a criterion for Out(G) to be finitely presented, as well as a necessary and sufficient condition for Out(G) to be finite. Consequences for hyperbolic groups are discussed.
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