an:07142608
Zbl 1456.20020
Guirardel, Vincent; Horbez, Camille
Algebraic laminations for free products and arational trees
EN
Algebr. Geom. Topol. 19, No. 5, 2283-2400 (2019).
1472-2747 1472-2739
2019
j
20E08 20E36 20F65 20F28 20E05
automorphisms of free products; algebraic laminations; group actions on trees; arational trees; geodesic currents; band complexes; rips machine
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 \textit{M. Bestvina} and \textit{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 \textit{M. Bestvina} and \textit{P. Reynolds} [Duke Math. J. 164, No. 11, 2213--2251 (2015; Zbl 1337.20040)] and \textit{U. Hamenstädt} [``The boundary of the free splitting graph and the free factor graph'', Preprint, \url{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.
V. A. Roman'kov (Omsk)
1348.20028; 1337.20040