Abstract
The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Osajda and Arzhantseva–Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.
References
[1] Vadim Alekseev and Martin Finn-Sell. Sofic boundaries of groups and coarse geometry of sofic approximations. Groups Geom. Dyn., 13(1):191–234, 2019.10.4171/GGD/482Search in Google Scholar
[2] S. Arnt. Fibred coarse embeddability of box spaces and proper isometric affine actions on Lp spaces. Bull. Belg. Math. Soc. Simon Stevin, 23(1):21–32, 2016.10.36045/bbms/1457560851Search in Google Scholar
[3] Goulnara Arzhantseva and Damian Osajda. Graphical small cancellation groups with the Haagerup property. preprint, arXiv:1404.6807, 2014.10.1142/S1793525315500144Search in Google Scholar
[4] Goulnara Arzhantseva and Romain Tessera. Relative expanders. Geom. Funct. Anal., 25(2):317–341, 2015.10.1007/s00039-015-0316-9Search in Google Scholar
[5] Uri Bader, Alex Furman, Tsachik Gelander, and Nicolas Monod. Property (T) and rigidity for actions on Banach spaces. Acta Math., 198(1):57–105, 2007.10.1007/s11511-007-0013-0Search in Google Scholar
[6] Keith Ball, Eric A. Carlen, and Elliott H. Lieb. Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math., 115(3):463–482, 1994.10.1007/BF01231769Search in Google Scholar
[7] Laurent Bartholdi and Anna Erschler. Ordering the space of finitely generated groups. Ann. Inst. Fourier (Grenoble), 65(5):2091–2144, 2015.10.5802/aif.2984Search in Google Scholar
[8] Helmut Behr. Arithmetic groups over function fields. I. A complete characterization of finitely generated and finitely presented arithmetic subgroups of reductive algebraic groups. J. Reine Angew. Math., 495:79–118, 1998.10.1515/crll.1998.023Search in Google Scholar
[9] Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s property (T), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008.10.1017/CBO9780511542749Search in Google Scholar
[10] Yoav Benyamini and Joram Lindenstrauss. Geometric nonlinear functional analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2000.Search in Google Scholar
[11] Mladen Bestvina, Ken Bromberg, and Koji Fujiwara. Constructing group actions on quasi-trees and applications to mapping class groups. Publ. Math. Inst. Hautes Études Sci., 122:1–64, 2015.10.1007/s10240-014-0067-4Search in Google Scholar
[12] Marc Bourdon. Un théorème de point fixe sur les espaces Lp. Publ. Mat., 56(2):375–392, 2012.10.5565/PUBLMAT_56212_05Search in Google Scholar
[13] Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.10.1007/978-3-662-12494-9Search in Google Scholar
[14] Sergei Buyalo and Viktor Schroeder. Elements of asymptotic geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2007.10.4171/036Search in Google Scholar
[15] Xiaoman Chen, Qin Wang, and Xianjin Wang. Characterization of the Haagerup property by fibred coarse embedding into Hilbert space. Bull. Lond. Math. Soc., 45(5):1091–1099, 2013.10.1112/blms/bdt045Search in Google Scholar
[16] Xiaoman Chen, Qin Wang, and Zhijie Wang. Fibred coarse embedding into non-positively curved manifolds and higher index problem. J. Funct. Anal., 267(11):4029–4065, 2014.10.1016/j.jfa.2014.10.004Search in Google Scholar
[17] Xiaoman Chen, Qin Wang, and Guoliang Yu. The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space. Adv. Math., 249:88–130, 2013.10.1016/j.aim.2013.09.003Search in Google Scholar
[18] Qingjin Cheng. Sphere equivalence, property H, and Banach expanders. Studia Math., 233(1):67–83, 2016.10.4064/sm8396-4-2016Search in Google Scholar
[19] Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette. Groups with the Haagerup property, volume 197 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2001. Gromov’s a-T-menability.10.1007/978-3-0348-0906-1Search in Google Scholar
[20] Giuliana Davidoff, Peter Sarnak, and Alain Valette. Elementary number theory, group theory, and Ramanujan graphs, volume 55 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2003.Search in Google Scholar
[21] Yves de Cornulier, Romain Tessera, and Alain Valette. Isometric group actions on Hilbert spaces: growth of cocycles. Geom. Funct. Anal., 17(3):770–792, 2007.10.1007/s00039-007-0604-0Search in Google Scholar
[22] Tim de Laat and Mikael de la Salle. Approximation properties for noncommutative Lp-spaces of high rank lattices and nonembeddability of expanders. J. Reine Angew. Math., 737:49–69, 2018.10.1515/crelle-2015-0043Search in Google Scholar
[23] Tim de Laat, Masato Mimura, and Mikael de la Salle. On strong property (T) and fixed point properties for Lie groups. Ann. Inst. Fourier (Grenoble), 66(5):1859–1893, 2016.10.5802/aif.3051Search in Google Scholar
[24] Martin Finn-Sell. Fibred coarse embeddings, a-T-menability and the coarse analogue of the Novikov conjecture. J. Funct. Anal., 267(10):3758–3782, 2014.10.1016/j.jfa.2014.09.012Search in Google Scholar
[25] R. I. Grigorchuk. Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat., 48(5):939–985, 1984.Search in Google Scholar
[26] M. Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., pages 1–295. Cambridge Univ. Press, Cambridge, 1993.Search in Google Scholar
[27] K. W. Gruenberg. Residual properties of infinite soluble groups. Proc. London Math. Soc. (3), 7:29–62, 1957.10.1112/plms/s3-7.1.29Search in Google Scholar
[28] Erik Guentner and Jerome Kaminker. Exactness and uniform embeddability of discrete groups. J. London Math. Soc. (2), 70(3):703–718, 2004.10.1112/S0024610704005897Search in Google Scholar
[29] Stefan Heinrich. Ultraproducts in Banach space theory. J. Reine Angew. Math., 313:72–104, 1980.10.1515/crll.1980.313.72Search in Google Scholar
[30] Hiroyasu Izeki and Shin Nayatani. Combinatorial harmonic maps and discrete-group actions on Hadamard spaces. Geom. Dedicata, 114:147–188, 2005.10.1007/s10711-004-1843-ySearch in Google Scholar
[31] Tom Kaiser. Combinatorial cost: a coarse setting. preprint, arXiv:1711.00413, 2017.Search in Google Scholar
[32] Takefumi Kondo. CAT(0) spaces and expanders. Math. Z., 271(1-2):343–355, 2012.10.1007/s00209-011-0866-ySearch in Google Scholar
[33] Nicholas J. Korevaar and Richard M. Schoen. Global existence theorems for harmonic maps to non-locally compact spaces. Comm. Anal. Geom., 5(2):333–387, 1997.10.4310/CAG.1997.v5.n2.a4Search in Google Scholar
[34] Vincent Lafforgue. Un renforcement de la propriété (T). Duke Math. J., 143(3):559–602, 2008.10.1215/00127094-2008-029Search in Google Scholar
[35] Vincent Lafforgue. Propriété (T) renforcée banachique et transformation de Fourier rapide. J. Topol. Anal., 1(3):191–206, 2009.10.1142/S1793525309000163Search in Google Scholar
[36] Benben Liao. Strong Banach property (T) for simple algebraic groups of higher rank. J. Topol. Anal., 6(1):75–105, 2014.10.1142/S1793525314500010Search in Google Scholar
[37] Alexander Lubotzky. Discrete groups, expanding graphs and invariant measures. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2010. With an appendix by Jonathan D. Rogawski, Reprint of the 1994 edition.10.1007/978-3-0346-0332-4_1Search in Google Scholar
[38] Manor Mendel and Assaf Naor. Expanders with respect to Hadamard spaces and random graphs. Duke Math. J., 164(8):1471–1548, 2015.10.1215/00127094-3119525Search in Google Scholar
[39] Masato Mimura. Sphere equivalence, Banach expanders, and extrapolation. Int. Math. Res. Not. IMRN, (12):4372–4391, 2015.10.1093/imrn/rnu075Search in Google Scholar
[40] Masato Mimura. An extreme counterexample to the Lubotzky–Weiss conjecture. preprint, arXiv:1809.08918v4, 2018.Search in Google Scholar
[41] Masato Mimura. Amenability versus non-exactness of dense subgroups of a compact group. J. London Math. Soc. (2), online published. Doi:10.1112/jlms.12229, 2019.10.1112/jlms.122292019Search in Google Scholar
[42] Masato Mimura, Narutaka Ozawa, Hiroki Sako, and Yuhei Suzuki. Group approximation in Cayley topology and coarse geometry, III: Geometric property (T). Algebr. Geom. Topol., 15(2):1067–1091, 2015.10.2140/agt.2015.15.1067Search in Google Scholar
[43] Masato Mimura and Hiroki Sako. Group approximation in Cayley topology and coarse geometry, part I: Coarse embeddings of amenable groups. preprint, arXiv:1310.4736v3. To appear in Journal of Topology and Analysis, online published. Doi: 10.1142/S1793525320500089, 2013.10.1142/S17935253205000892013Search in Google Scholar
[44] Assaf Naor. Comparison of metric spectral gaps. Anal. Geom. Metr. Spaces, 2:1–52, 2014.10.2478/agms-2014-0001Search in Google Scholar
[45] Assaf Naor and Yuval Peres. Lp compression, traveling salesmen, and stable walks. Duke Math. J., 157(1):53–108, 2011.10.1215/00127094-2011-002Search in Google Scholar
[46] Assaf Naor and Lior Silberman. Poincaré inequalities, embeddings, and wild groups. Compos. Math., 147(5):1546–1572, 2011.10.1112/S0010437X11005343Search in Google Scholar
[47] Piotr W. Nowak and Guoliang Yu. Large scale geometry. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2012.Search in Google Scholar
[48] Damian Osajda. Small cancellation labellings of some infinite graphs and applications. preprint, arXiv:1406.5015, 2014.Search in Google Scholar
[49] Damian Osajda. Residually finite non-exact groups. Geom. Funct. Anal., 28(2):509–517, 2018.10.1007/s00039-018-0432-4Search in Google Scholar
[50] M. I. Ostrovskii. Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Proc. Amer. Math. Soc., 140(8):2721–2730, 2012.10.1090/S0002-9939-2011-11272-3Search in Google Scholar
[51] Narutaka Ozawa. A note on non-amenability of B(lp) for p = 1, 2. Internat. J. Math., 15(6):557–565, 2004.10.1142/S0129167X04002430Search in Google Scholar
[52] Thibault Pillon. Coarse amenability at infinity. preprint, arXiv:1812.11745, 2018.Search in Google Scholar
[53] Gilles Pisier. Martingales with values in uniformly convex spaces. Israel J. Math., 20(3-4):326–350, 1975.10.1007/BF02760337Search in Google Scholar
[54] Yves Raynaud. On ultrapowers of non commutative Lp spaces. J. Operator Theory, 48(1):41–68, 2002.Search in Google Scholar
[55] Damian Sawicki. Warped cones, (non-)rigidity, and piecewise properties, with a joint appendix with Dawid Kielak. Proc. London Math. Soc. (3), 118:753–786, 2019.10.1112/plms.12192Search in Google Scholar
[56] Atle Selberg. On the estimation of Fourier coefficients of modular forms. In Proc. Sympos. Pure Math., Vol. VIII, pages 1–15. Amer. Math. Soc., Providence, R.I., 1965.10.1090/pspum/008/0182610Search in Google Scholar
[57] Yehuda Shalom. Rigidity of commensurators and irreducible lattices. Invent. Math., 141(1):1–54, 2000.10.1007/s002220000064Search in Google Scholar
[58] Yves Stalder. Fixed point properties in the space of marked groups. In Limits of graphs in group theory and computer science, pages 171–182. EPFL Press, Lausanne, 2009.Search in Google Scholar
[59] Nicole Tomczak-Jaegermann. Banach-Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.Search in Google Scholar
[60] Tetsu Toyoda. Continuity of a certain invariant of a measure on a CAT(0) space. Nihonkai Math. J., 20(2):85–97, 2009.Search in Google Scholar
[61] Tetsu Toyoda. CAT(0) spaces on which a certain type of singularity is bounded. Kodai Math. J., 33(3):398–415, 2010.10.2996/kmj/1288962550Search in Google Scholar
[62] A. M. Vershik and E. I. Gordon. Groups that are locally embeddable in the class of finite groups. Algebra i Analiz, 9(1):71–97, 1997.Search in Google Scholar
[63] Guoliang Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math., 139(1):201–240, 2000.10.1007/s002229900032Search in Google Scholar
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