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Outer automorphisms of hyperbolic groups and small actions on \(\mathbb{R}\)- trees. (English) Zbl 0804.57002

Arboreal group theory, Proc. Workshop, Berkeley/CA (USA) 1988, Publ., Math. Sci. Res. Inst. 19, 331-343 (1991).
An \(\mathbb{R}\)-tree is a metric space such that any two points are connected by a unique arc isometric to an interval of the reals. Let \(\Gamma\) be a finitely generated (word) hyperbolic group acting on an \(\mathbb{R}\)-tree. An edge stabilizer is a subgroup of \(\Gamma\) which stabilizes one of the aforementioned unique arcs connecting points of the tree. An edge stabilizer (or any group for that matter) is said to be almost cyclic if every finitely generated subgroup has a cyclic subgroup of finite index. The main result of the paper then connects the structure of \(\Gamma\) to the kind of action \(\Gamma\) can have on an \(\mathbb{R}\)-tree. Precisely, the author shows that if the outer automorphism group of \(\Gamma\) is infinite, then there is an isometric action of \(\Gamma\) on an \(\mathbb{R}\)- tree with almost cyclic edge stabilizers and without any global fixed point.
For the entire collection see [Zbl 0744.00026].

MSC:

57M07 Topological methods in group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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