\(\mathbb{R}\)-trees in topology, geometry, and group theory (Q2776330)

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. NEWLINENEWLINENEWLINEFollowing 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. NEWLINENEWLINENEWLINEThe 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. NEWLINENEWLINENEWLINEIn 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. NEWLINENEWLINENEWLINEThe 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.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].