Characterising quasi-isometries of the free group (Q6661617)

The set of quasi-isometries of a metric space is, in a natural way a group, about which, in general, very little is known. No effective methods are known to describe all quasi-isometries, except in some cases where quasi-isometric rigidity is known. Two notable exceptions to this state of affairs are known so far: Baumslag-Solitar groups, which quasi-isometries are described in [\textit{K. Whyte}, Geom. Funct. Anal. 11, No. 6, 1327--1343 (2001; Zbl 1004.20024)] and 3-dimensional solvable Lie groups, which have been studied by \textit{A. Eskin} et al. in [Ann. Math. (2) 176, No. 1, 221--260 (2012; Zbl 1264.22005); Ann. Math. (2) 177, No. 3, 869--910 (2013; Zbl 1398.22012)].\N\NIn the paper under review, the authors introduce the notion of mixed subtree quasi-isometries (for a precise definition, see Section 3), which are self-quasi-isometries of regular trees built in a specific inductive way. The main result is.\N\NTheorem 1.1: Let \(T\) be a regular tree of degree at least 3, rooted at \(v_{0}\). Let \(f: T \rightarrow T\) be a \(C\)-quasi-isometry such that \(f(v_{0})=v_{0}\). Then there is a constant \(D\) only depending on \(C\) and a \(D\)-deep mixed subtree quasi-isometry \(g: T \rightarrow T\) such that \(f\) and \(g\) are at bounded distance from each other.\N\NSince regular trees of degree at least \(3\) and non-elementary free groups are quasi-isometric, the theorem above describes quasi-isometries of the free group \(\mathbb{F}_{2}\). This case is therefore added to the two already known cases mentioned at the beginning.\N\NMixed-subtree quasi-isometries seem to be an useful tool to construct quasi-isometries with certain desired properties. For example, the authors, in [J. Theor. Probab. 37, No. 3, 2330--2351 (2024; Zbl 07900851)], use this technique to build a self-quasi-isometry of \(\mathbb{F}_{2}\) with the property that the push-forward of a simple random walk by this quasi-isometry does not have a well-defined drift.