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Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture. (English) Zbl 1017.54017
A metric space \(X\) is UDBG (uniformly discrete of bounded geometry) if for some \(\varepsilon>0\) any two points \(x\), \(x'\in X\) are separated by a distance at least \(\varepsilon\), and for any \(r>0\) there is a bound \(N_r\) of the cardinality of any \(r\)-ball in \(X\). The main result of the paper is the following: If \(f:X\to Y\) is a quasi-isometry between UDBG spaces, then there is a bi-Lipschitz map at bounded distance from \(f\) if and only if \(f_\ast([X])=[Y]\), where \([X]\) and \([Y]\) are the fundamental classes in \(H_0^{uf}\).
The uniformly finite homology groups \(H_\ast^{uf}(X)\) were introduced by J. Block and S. Weinberger in [J. Am. Math. Soc. 5, 907-918 (1992; Zbl 0780.53031)], where they proved that \(X\) is nonamenable if and only if \(H_0^{uf}(X)=0\). Thus the theorem above generalizes an earlier result of P. Papasoglu [Geom. Dedicata 54, 301-306 (1995; Zbl 0836.05018)] on bi-Lipschitz equivalence of homogeneous trees. As an application, the author works out the quasi-isometric classification of free products of (any finite number of) surface groups. In particular, it is shown that neither the sign of the Euler characteristic nor ratios of \(l^2\)-Betti numbers are quasi-isometric invariant, which answers a question of Gromov. Another application of the main result is the following theorem (which is called a geometric version of the von Neumann conjecture):
An UDBG space \(Z\) is nonamenable if and only if \(Z\) admits a free action by a free group on two generators by bi-Lipschitz maps at bounded distance from the identity.
As a consequence, it is shown that the vanishing of the 0-dimensional \(l^p\)-homology for any \(p\in(1,\infty]\) is equivalent to nonamenability. The case \(p=\infty\) is the result by J. Block and S. Weinberger mentioned above.

MSC:
54E35 Metric spaces, metrizability
20F65 Geometric group theory
53C99 Global differential geometry
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
55N35 Other homology theories in algebraic topology
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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