Abstract
We prove by elementary means a regularity theorem for quasi-isometries of T x ℝn (where T denotes an infinite tree), and of many other metric spaces with similar combinatorial properties, e.g. Cayley graphs of Baumslag–Solitar groups. For quasi-isometries of T x ℝn, it states that the image of {x} x ℝn (xεT) is uniformly close to {y} x ℝn for some yεT, and there is a well-defined surjection \(QI(T \times \mathbb{R}^n ) \to QI(T)\). Even stronger, the image of a quasi-isometric embedding of ℝn+1 in T x ℝn is close to (a geodesic in T)T)x ℝn.
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Souche, E., Wiest, B. An Elementary Approach to Quasi-Isometries of Tree x ℝn . Geometriae Dedicata 95, 87–102 (2002). https://doi.org/10.1023/A:1021241616769
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DOI: https://doi.org/10.1023/A:1021241616769