# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Jonathan Block and Shmuel Weinberger, Aperiodic tilings, positive scalar curvature and amenability of spaces , J. Amer. Math. Soc. 5 (1992), no. 4, 907-918. JSTOR: · Zbl 0780.53031 · doi:10.2307/2152713 · links.jstor.org [2] Jonathan Block and Shmuel Weinberger, Large scale homology theories and geometry , Geometric topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 522-569. · Zbl 0898.55006 [3] D. Burago and B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps , Geom. Funct. Anal. 8 (1998), no. 2, 273-282. · Zbl 0902.26004 · doi:10.1007/s000390050056 [4] Jeff Cheeger and Mikhael Gromov, $$L_ 2$$-cohomology and group cohomology , Topology 25 (1986), no. 2, 189-215. · Zbl 0597.57020 · doi:10.1016/0040-9383(86)90039-X [5] Gerald B. Folland, Real analysis , Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1984, Modern Techniques and Their Applications. · Zbl 0549.28001 [6] S. M. Gersten, Bounded cocycles and combings of groups , Internat. J. Algebra Comput. 2 (1992), no. 3, 307-326. · Zbl 0762.57001 · doi:10.1142/S0218196792000190 [7] S. Gersten, A homological characterization of hyperbolic groups , preprint, 1998. [8] Jack E. Graver and Mark E. Watkins, Combinatorics with emphasis on the theory of graphs , Springer-Verlag, New York, 1977, Grad. Texts in Math., 54. · Zbl 0367.05001 [9] M. Gromov, “Asymptotic invariants of infinite groups” , Geometric Group Theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295. · Zbl 0841.20039 [10] Paul R. Halmos, Naive set theory , Springer-Verlag, New York, 1974, Undergrad. Texts Math. · Zbl 0287.04001 [11] C. T. McMullen, Lipschitz maps and nets in Euclidean space , Geom. Funct. Anal. 8 (1998), no. 2, 304-314. · Zbl 0941.37030 · doi:10.1007/s000390050058 [12] A. Ju. Olšanskiĭ, On the question of the existence of an invariant mean on a group , Uspekhi Mat. Nauk 35 (1980), no. 4(214), 199-200, (in Russian); English transl. in Russian Math. Surveys 35, no. 4 (1980), 180-181. · Zbl 0452.20032 [13] P. Papasoglu, Homogeneous trees are bi-Lipschitz equivalent , Geom. Dedicata 54 (1995), no. 3, 301-306. · Zbl 0836.05018 · doi:10.1007/BF01265344 [14] Jean-Pierre Serre, Trees , Springer-Verlag, Berlin, 1980. · Zbl 0548.20018 [15] J. Tits, Free subgroups in linear groups , J. Algebra 20 (1972), 250-270. · Zbl 0236.20032 · doi:10.1016/0021-8693(72)90058-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.