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$$\mathbb{R}$$-trees in topology, geometry, and group theory. (English) Zbl 0998.57003
Daverman, R. J. (ed.) et al., Handbook of geometric topology. Amsterdam: Elsevier. 55-91 (2002).
This paper is a survey of the theory and applications of $$\mathbb{R}$$-trees. Many proofs are sketched and key ideas are nicely presented. The paper starts with examples and basic properties of $$\mathbb{R}$$-trees, and an insightful discussion of how $$\mathbb{R}$$-trees arise in geometry and group theory.
Following is a discussion of measured laminations on $$2$$-complexes. If $$G$$ is the fundamental group of a finite $$2$$-complex $$K$$, then any isometric minimal nontrivial action of $$G$$ on an $$\mathbb{R}$$-tree gives rise to a measured lamination on $$K$$. For simplicial trees this idea goes back to Stallings and was used extensively by Dunwoody.
The heart of the paper is describing the “Rips machine” which is an algorithm that takes an input a finite $$2$$-complex equipped with transversely measured lamination (or more precisely, a band complex), and puts it in a “normal form”. In the normal form the lamination becomes the disjoint union of finitely many sub-laminations of one of the following types: simplicial, surface, toral, and thin band complex.
In particular, the Rips machine yields a classification of stable actions of finitely presented groups on $$\mathbb{R}$$-trees, which was developed by Bestvina and Feighn following the breakthrough of Rips.
The list of applications includes compactifying spaces of hyperbolic structures, studying endomorphisms of word-hyperbolic groups by Rips and Sela, a new proof of the Bestvina-Handle theorem stating that if $$f$$ is an automorphism of a rank $$n$$ free group, then the subgroup fixed by $$f$$ has rank at most $$n$$, and a theorem of Bowditch and Swarup stating that the boundary of a word hyperbolic one-ended group has no cut points.
For the entire collection see [Zbl 0977.00029].

##### MSC:
 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 20E08 Groups acting on trees 57M07 Topological methods in group theory