## Coarse differentiation of quasi-isometries. I: Spaces not quasi-isometric to Cayley graphs.(English)Zbl 1264.22005

Given a metric space $$X$$ denote by $$\operatorname{QI}(X)$$ the group of quasi-isometries of $$X$$ modulo those of bounded distance from the identity. The paper under review computes the group $$\operatorname{QI}(X)$$ for various horocyclic products of negatively-curved spaces and applies these results to establish important new quasi-isometric rigidity results. The proofs are based on the method of coarse differentiation introduced by the authors in [Pure Appl. Math. Q. 3, No. 4, 927–947 (2007; Zbl 1167.22007)].
Recall that given two negatively-curved spaces $$X_1$$, $$X_2$$ with respective Busemann functions $$f_1$$, $$f_2$$, their horocyclic product is the subset $X := \{(x_1, x_2) \in X_1 \times X_2\mid f_1(x_1) = -f_2(x_2)\}$ of the product $$X_1 \times X_2$$. Such spaces come equipped with a natural height function $$h := f_1 = -f_2$$. The authors consider specifically the family of Diestel-Leader graphs DL$$(m,n)$$, which are horocyclic products of two regular trees of valencies $$m+1$$ and $$n+1$$, respectively, and a family of $$3$$-dimensional solvable Lie groups Sol$$(m,n)$$, which are isomorphic to horocyclic products of two planes of constant negative curvatures depending on the parameters $$m$$ and $$n$$.
Quasi-isometries of these spaces which coarsely preserve the height function have been classified by Farb and Mosher. They are all essentially equivalent to restrictions of products of quasi-isometries of the two factors, hence correspond to pairs of bi-Lipschitz maps of the corresponding pointed boundaries. The main technical result of the paper under review greatly improves on this result by showing that actually every quasi-isometry is at bounded distance from a coarsely height-preserving one, hence of the above product form (Theorem 2.1/Theorem 2.3). In the case of distinct curvatures $$m \neq n$$, this implies $\operatorname{QI}(\operatorname{Sol}(m,n)) \cong \operatorname{Bilip}(\mathbb R)^2, \quad \operatorname{QI}(\operatorname{DL}(m,n)) \cong \operatorname{Bilip}(\mathbb Q_m) \times \operatorname{Bilip}(\mathbb Q_n),$ whereas, in the case of equal curvature $$m=n$$, there is an additional $$\mathbb Z/2\mathbb Z$$-factor corresponding to the flip. Actually, only the distinct curvature case is completely established in the present paper, whereas parts of the proof in the case of equal curvature are deferred to a sequel.
The identifications of the above $$\operatorname{QI}$$-groups have a number of striking applications. Firstly, if $$m \neq n$$, then neither $$\operatorname{Sol}(m,n)$$ nor DL$$(m,n)$$ are quasi-isometric to Cayley graphs of finitely-generated groups (Theorem 1.2/Theorem 1.4), which is the result alluded to in the title. Secondly, the authors determine completely which of the spaces $$\operatorname{Sol}(m,n)$$, respectively, $$\operatorname{DL}(m,n)$$, are quasi-isometric to each other, assuming $$m \neq n$$. For example, $$\operatorname{Sol}(m,n)$$ is quasi-isometric to $$\operatorname{Sol}(m',n')$$ if and only if $$m/m' = n/n'$$ (Theorem 1.3). Both results have counterparts in the equal curvature case (the former in terms of a quasi-isometric rigidity statement), but these are deferred to the sequel. To deduce the above results from the knowledge of the QI-groups, the authors rely heavily on previous results concerning quasi-isometric quasi-actions on trees and hyperbolic spaces.
The starting point for the proof of the main technical result is a coarse version of Rademacher’s theorem (Theorem 4.3), in which the authors show that quasi-isometries of the spaces in question can be coarsely differentiated on a set of large measure. Using this theorem, the authors deduce that on a set of large measure any given quasi-isometry looks locally like standard product quasi-isometries. The main technical step (Step II) is to show that the local maps preserve the up-direction, which is necessary to assemble the local information. This is the only step where the difference between the distinct curvature case and the equal curvature case becomes relevant.
The results demonstrate the power of the method of coarse differentiation to establish new rigidity results, in particular for solvable groups. It would be interesting to know which of the arguments generalize to more general horocyclic products.

### MSC:

 22E25 Nilpotent and solvable Lie groups 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 20F65 Geometric group theory 22E40 Discrete subgroups of Lie groups 53C24 Rigidity results 05C05 Trees 05C76 Graph operations (line graphs, products, etc.)

Zbl 1167.22007
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### References:

 [1] S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, ”Affine approximation of Lipschitz functions and nonlinear quotients,” Geom. Funct. Anal., vol. 9, iss. 6, pp. 1092-1127, 1999. · Zbl 0954.46014 [2] I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, ”Group-invariant percolation on graphs,” Geom. Funct. Anal., vol. 9, iss. 1, pp. 29-66, 1999. · Zbl 0924.43002 [3] Y. Binyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Providence, RI: Amer. Math. Soc., 2000, vol. 48. · Zbl 0946.46002 [4] J. Bourgain, ”Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms,” in Geometrical Aspects of Functional Analysis (1985/86), New York: Springer-Verlag, 1987, vol. 1267, pp. 157-167. · Zbl 0633.46018 [6] P. de la Harpe, Topics in Geometric Group Theory, Chicago, IL: University of Chicago Press, 2000. · Zbl 0965.20025 [7] R. Diestel and I. Leader, ”A conjecture concerning a limit of non-Cayley graphs,” J. Algebraic Combin., vol. 14, iss. 1, pp. 17-25, 2001. · Zbl 0985.05020 [8] G. Elek and G. Tardos, ”On roughly transitive amenable graphs and harmonic Dirichlet functions,” Proc. Amer. Math. Soc., vol. 128, iss. 8, pp. 2479-2485, 2000. · Zbl 0948.58017 [9] A. Eskin, ”Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces,” J. Amer. Math. Soc., vol. 11, iss. 2, pp. 321-361, 1998. · Zbl 0885.22017 [10] A. Eskin and B. Farb, ”Quasi-flats and rigidity in higher rank symmetric spaces,” J. Amer. Math. Soc., vol. 10, iss. 3, pp. 653-692, 1997. · Zbl 0893.22004 [11] A. Eskin, D. Fisher, and K. Whyte, ”Quasi-isometries and rigidity of solvable groups,” Pure Appl. Math. Q., vol. 3, iss. 4, part 1, pp. 927-947, 2007. · Zbl 1167.22007 [12] A. Eskin, D. Fisher, and K. Whyte, Coarse differentiation of quasi-isometries II; Rigidity for lattices in $$\mathrm{Sol}$$ and Lamplighter groups. · Zbl 1398.22012 [13] B. Farb and R. Schwartz, ”The large-scale geometry of Hilbert modular groups,” J. Differential Geom., vol. 44, iss. 3, pp. 435-478, 1996. · Zbl 0871.11035 [14] B. Farb, ”The quasi-isometry classification of lattices in semisimple Lie groups,” Math. Res. Lett., vol. 4, iss. 5, pp. 705-717, 1997. · Zbl 0889.22010 [15] B. Farb and L. Mosher, ”A rigidity theorem for the solvable Baumslag-Solitar groups,” Invent. Math., vol. 131, iss. 2, pp. 419-451, 1998. · Zbl 0937.22003 [16] B. Farb and L. Mosher, ”Quasi-isometric rigidity for the solvable Baumslag-Solitar groups. II,” Invent. Math., vol. 137, iss. 3, pp. 613-649, 1999. · Zbl 0931.20035 [17] B. Farb and L. Mosher, ”On the asymptotic geometry of abelian-by-cyclic groups,” Acta Math., vol. 184, iss. 2, pp. 145-202, 2000. · Zbl 0982.20026 [18] B. Farb and L. Mosher, ”Problems on the geometry of finitely generated solvable groups,” in Crystallographic Groups and their Generalizations, Providence, RI: Amer. Math. Soc., 2000, vol. 262, pp. 121-134. · Zbl 0983.20038 [19] M. Gromov, ”Groups of polynomial growth and expanding maps,” Inst. Hautes Études Sci. Publ. Math., vol. 53, pp. 53-73, 1981. · Zbl 0474.20018 [20] M. Gromov, ”Infinite groups as geometric objects,” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Warsaw, 1984, pp. 385-392. · Zbl 0586.20016 [21] M. Gromov, ”Asymptotic invariants of infinite groups,” in Geometric Group Theory, Vol. 2, Cambridge: Cambridge Univ. Press, 1993, vol. 182, pp. 1-295. · Zbl 0841.20039 [22] J. Heinonen, Lectures on Analysis on Metric Spaces, New York: Springer-Verlag, 2001. · Zbl 0985.46008 [23] A. Hinkkanen, ”Uniformly quasisymmetric groups,” Proc. London Math. Soc., vol. 51, iss. 2, pp. 318-338, 1985. · Zbl 0574.30022 [24] W. B. Johnson, J. Lindenstrauss, and G. Schechtman, ”Banach spaces determined by their uniform structures,” Geom. Funct. Anal., vol. 6, iss. 3, pp. 430-470, 1996. · Zbl 0864.46008 [25] B. Kleiner and B. Leeb, ”Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings,” Inst. Hautes Études Sci. Publ. Math., vol. 86, pp. 115-197, 1997. · Zbl 0910.53035 [26] B. Kleiner, Personal communication. [27] J. Matouvsek, ”On embedding trees into uniformly convex Banach spaces,” Israel J. Math., vol. 114, pp. 221-237, 1999. · Zbl 0948.46011 [28] R. Moeller, Letter to W. Woess, 2011. [29] L. Mosher, M. Sageev, and K. Whyte, ”Quasi-actions on trees. I. Bounded valence,” Ann. of Math., vol. 158, iss. 1, pp. 115-164, 2003. · Zbl 1038.20016 [30] P. Pansu, ”Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un,” Ann. of Math., vol. 129, iss. 1, pp. 1-60, 1989. · Zbl 0678.53042 [31] P. Pansu, ”Dimension conforme et sphère à l’infini des variétés à courbure négative,” Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 14, iss. 2, pp. 177-212, 1989. · Zbl 0722.53028 [32] Y. Peres, P. Gábor, and A. Scolnicov, ”Critical percolation on certain nonunimodular graphs,” New York J. Math., vol. 12, pp. 1-18 (electronic), 2006. · Zbl 1103.60079 [33] D. Preiss, ”Differentiability of Lipschitz functions on Banach spaces,” J. Funct. Anal., vol. 91, iss. 2, pp. 312-345, 1990. · Zbl 0711.46036 [34] R. E. Schwartz, ”The quasi-isometry classification of rank one lattices,” Inst. Hautes Études Sci. Publ. Math., vol. 82, pp. 133-168, 1995. · Zbl 0852.22010 [35] R. E. Schwartz, ”Quasi-isometric rigidity and Diophantine approximation,” Acta Math., vol. 177, iss. 1, pp. 75-112, 1996. · Zbl 1029.11038 [36] Y. Shalom, ”Harmonic analysis, cohomology, and the large-scale geometry of amenable groups,” Acta Math., vol. 192, iss. 2, pp. 119-185, 2004. · Zbl 1064.43004 [37] P. M. Soardi and W. Woess, ”Amenability, unimodularity, and the spectral radius of random walks on infinite graphs,” Math. Z., vol. 205, iss. 3, pp. 471-486, 1990. · Zbl 0693.43001 [38] W. Woess, ”Topological groups and infinite graphs,” Discrete Math., vol. 95, iss. 1-3, pp. 373-384, 1991. · Zbl 0757.05060 [39] W. Woess, ”Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions,” Combin. Probab. Comput., vol. 14, iss. 3, pp. 415-433, 2005. · Zbl 1066.05075 [40] K. Wortman, ”A finitely presented solvable group with a small quasi-isometry group,” Michigan Math. J., vol. 55, iss. 1, pp. 3-24, 2007. · Zbl 1137.20034
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