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Rigidity of warped cones and coarse geometry of expanders. (English) Zbl 1440.53047

Summary: We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions are finite covers of conjugate actions. As a consequence, we produce continuous families of non-quasi-isometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental groups.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C24 Rigidity results
51F30 Lipschitz and coarse geometry of metric spaces
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[1] Barcelo, H.; Capraro, V.; White, J. A., Discrete homology theory for metric spaces, Bull. Lond. Math. Soc., 46, 5, 889-905 (2014) · Zbl 1308.55004
[2] Benoist, Y.; de Saxcé, N., A spectral gap theorem in simple Lie groups, Invent. Math., 205, 2, 337-361 (2016) · Zbl 1357.22003
[3] Bestvina, M., Degenerations of the hyperbolic space, Duke Math. J., 56, 1, 143-161 (1988) · Zbl 0652.57009
[4] Borel, A., On free subgroups of semisimple groups, Enseign. Math. (2), 29, 1-2, 151-164 (1983) · Zbl 0533.22009
[5] Bourgain, J.; Gamburd, A., On the spectral gap for finitely-generated subgroups of \(SU(2)\), Invent. Math., 171, 1, 83-121 (2008) · Zbl 1135.22010
[6] Bourgain, J.; Gamburd, A., A spectral gap theorem in \(SU(d)\), J. Eur. Math. Soc. (JEMS), 14, 5, 1455-1511 (2012) · Zbl 1254.43010
[7] Bridson, M. R.; Haefliger, A., Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319 (1999), Springer-Verlag: Springer-Verlag Berlin · Zbl 0988.53001
[8] Canary, R. D., Dynamics on character varieties: a survey, (Handbook of Group Actions, II (2015)), 175-200
[9] Culler, M.; Morgan, J. W., Group actions on R-trees, Proc. Lond. Math. Soc. (3), 55, 3, 571-604 (1987) · Zbl 0658.20021
[10] de Laat, T.; Vigolo, F., Superexpanders from group actions on compact manifolds, Geom. Dedicata (2017), in press · Zbl 07065313
[11] Delabie, T.; Khukhro, A., Coarse fundamental groups and box spaces (2017), preprint
[12] Drinfeld, V. G., Finitely-additive measures on \(S^2\) and \(S^3\), invariant with respect to rotations, Funktsional. Anal. i Prilozhen., 18, 3, 77 (1984)
[13] Drutu, C.; Nowak, P., Kazhdan projections, random walks and ergodic theorems, Crelle’s J. (2015), in press
[14] Fisher, D.; Whyte, K., When is a group action determined by its orbit structure?, Geom. Funct. Anal., 13, 6, 1189-1200 (2003) · Zbl 1055.20031
[15] Gabber, O.; Galil, Z., Explicit constructions of linear-sized superconcentrators, J. Comput. System Sci., 22, 3, 407-420 (1981), (special issue dedicated to Michael Machtey) · Zbl 0487.05045
[16] Gamburd, A.; Jakobson, D.; Sarnak, P., Spectra of elements in the group ring of \(SU(2)\), J. Eur. Math. Soc. (JEMS), 1, 1, 51-85 (1999) · Zbl 0916.22009
[17] Hume, D., A continuum of expanders, Fund. Math., 238, 2, 143-152 (2017) · Zbl 1400.05111
[18] Khukhro, A.; Valette, A., Expanders and box spaces, Adv. Math., 314, 806-834 (2017) · Zbl 1426.20015
[19] Kim, H. J., Coarse equivalences between warped cones, Geom. Dedicata, 120, 19-35 (2006) · Zbl 1097.53021
[20] Knapp, A. W., Lie Groups Beyond an Introduction, Progr. Math., vol. 140 (2002), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1075.22501
[21] Lafforgue, V., Un renforcement de la propriété (T), Duke Math. J., 143, 3, 559-602 (2008) · Zbl 1158.46049
[22] Liao, B., Strong Banach property (T) for simple algebraic groups of higher rank, J. Topol. Anal., 6, 1, 75-105 (2014) · Zbl 1291.22010
[23] Margulis, G. A., Explicit constructions of expanders, Problemy Peredachi Informatsii, 9, 4, 71-80 (1973) · Zbl 0312.22011
[24] Margulis, G. A., Some remarks on invariant means, Monatsh. Math., 90, 3, 233-235 (1980) · Zbl 0425.43001
[25] Mendel, M.; Naor, A., Nonlinear spectral calculus and super-expanders, Publ. Math. Inst. Hautes Études Sci., 119, 1-95 (2014) · Zbl 1306.46021
[26] Mendel, M.; Naor, A., Expanders with respect to Hadamard spaces and random graphs, Duke Math. J., 164, 8, 1471-1548 (2015) · Zbl 1316.05109
[27] Nisnewitsch, V. L., Über Gruppen, die durch Matrizen über einem kommutativen Feld isomorph darstellbar sind, Rec. Math. [Mat. Sb.] N.S., 8, 50, 395-403 (1940) · Zbl 0024.25303
[28] Nowak, P. W.; Sawicki, D., Warped cones and spectral gaps, Proc. Amer. Math. Soc., 145, 2, 817-823 (2017) · Zbl 1368.46025
[29] Ostrovskii, M. I., Bilipschitz and coarse embeddings into Banach spaces, (Metric Embeddings. Metric Embeddings, De Gruyter Studies in Mathematics, vol. 49 (2013), De Gruyter: De Gruyter Berlin) · Zbl 1279.46001
[30] Paulin, F., Outer automorphisms of hyperbolic groups and small actions on R-trees, (Arboreal Group Theory. Arboreal Group Theory, Berkeley, CA, 1988. Arboreal Group Theory. Arboreal Group Theory, Berkeley, CA, 1988, Math. Sci. Res. Inst. Publ., vol. 19 (1991), Springer: Springer New York), 331-343 · Zbl 0804.57002
[31] Roe, J., Warped cones and property A, Geom. Topol., 9, 163-178 (2005) · Zbl 1082.53036
[32] Sawicki, D., Super-expanders and warped cones (2016), preprint
[33] Sawicki, D., Warped cones, (non-)rigidity, and piecewise properties, with a joint appendix with Dawid Kielak (2017), preprint
[34] D. Sawicki, Warped cones violating the coarse Baum-Connes conjecture, preprint, 2017.; D. Sawicki, Warped cones violating the coarse Baum-Connes conjecture, preprint, 2017.
[35] Serre, J.-P., Trees (1980), Springer-Verlag: Springer-Verlag Berlin-New York, translated from the French by John Stillwell
[36] Shalen, P. B., Linear representations of certain amalgamated products, J. Pure Appl. Algebra, 15, 2, 187-197 (1979) · Zbl 0401.20024
[37] Sullivan, D., For \(n > 3\) there is only one finitely additive rotationally invariant measure on the \(n\)-sphere defined on all Lebesgue measurable subsets, Bull. Amer. Math. Soc. (N.S.), 4, 1, 121-123 (1981) · Zbl 0459.28009
[38] Vigolo, F., Fundamental groups as limits of discrete fundamental groups, Bull. Lond. Math. Soc., 50, 5, 801-810 (2018) · Zbl 1436.20070
[39] Vigolo, F., Measure expanding actions, expanders and warped cones, Trans. Amer. Math. Soc., 371, 3, 1951-1979 (2019) · Zbl 1402.05205
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