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Algebraic laminations for free products and arational trees. (English) Zbl 1456.20020
In analogy to curve complexes used to study mapping class groups of surfaces, the free factor graph of a free group \(F_n\) has recently turned to be fruitful in the study of Out(\(F_n\)). It is Gromov hyperbolic, as was proved by M. Bestvina and M. Feighn [Adv. Math. 256, 104–155 (2014; Zbl 1348.20028)], and the action of an automorphism of \(F_n\) is loxodromic if and only if it is fully irreducible. Its Gromov boundary was described by M. Bestvina and P. Reynolds [Duke Math. J. 164, No. 11, 2213–2251 (2015; Zbl 1337.20040)] and U. Hamenstädt [“The boundary of the free splitting graph and the free factor graph”, Preprint, arXiv:1211.1630] as the set of equivalence classes of arational trees.
The main goal of the paper under review is to extend the theory of algebraic laminations to the context of free products. A key point for this intended application says that if two trees have a leaf in common in their dual laminations, and if one of the trees is arational and relatively free, then they are equivariantly homeomorphic.

MSC:
20E08 Groups acting on trees
20E36 Automorphisms of infinite groups
20F65 Geometric group theory
20F28 Automorphism groups of groups
20E05 Free nonabelian groups
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