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Constructing free actions on \(\mathbf R\)-trees. (English) Zbl 0794.57001
From the author’s introduction: “This paper describes two constructions leading to free group actions on \(R\)-trees. In the first one we start with an arbitrary action \((G,T)\), and we construct actions of certain quotients \(G/H\) on quotient \(R\)-trees \(\widehat{T/H}\). Among these actions, there is a ‘largest’ free one, so that we can associate a free action to \((G,T)\) in a canonical way. In the second construction, we use pseudogroups of rotations of the circle constructed in [G. Levitt, Invent. Math. 113, No. 3, 633-670 (1993; Zbl 0791.58055)] to get free nonsimplicial actions of the free group of rank 3. The translation lengths of the generators may be any triple of positive, rationally independent numbers. Both constructions use measured foliations.” Concerning the latter examples: “Using automorphisms of free groups, Bestvina and Handel have constructed a countable family of nonsimplicial free actions of free groups. The length function associated to any of their actions takes its values in a finite algebraic extension of \(Q\).”
While developing these constructions, the author proves a number of interesting technical results, such as criteria for recognizing when the quotient of an action on an \(R\)-tree is again an \(R\)-tree, and a general construction of a measured foliation whose leaf space made Hausdorff is isometric to \(\widehat{T/H}\).

MSC:
57M07 Topological methods in group theory
57M60 Group actions on manifolds and cell complexes in low dimensions
20E08 Groups acting on trees
57R30 Foliations in differential topology; geometric theory
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