An Elementary Approach to Quasi-Isometries of Tree x ℝn

Author: Souche E.   Wiest B.  

Publisher: Springer Publishing Company

ISSN: 0046-5755

Source: Geometriae Dedicata, Vol.95, Iss.1, 2002-12, pp. : 87-102

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Abstract

We prove by elementary means a regularity theorem for quasi-isometries of Ttimes {open R}^n (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 Ttimes {open R}^n, it states that the image of {x}times {open R}^n (xT) is uniformly close to {y}times {open R}^n for some yT, and there is a well-defined surjection QI(Ttimes {open R}^n) to QI(T,). Even stronger, the image of a quasi-isometric embedding of {open R}^{n+1} in Ttimes {open R}^n is close to (a geodesic in T)times {open R}^n.