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UNIFORM KAZHDAN CONSTANTS AND PARADOXES OF THE AFFINE PLANE

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Abstract

Let G = SL(2, ℤ) ⋉ ℤ2 and H = SL(2, ℤ). We prove that the action G ↷ ℝ2 is uniformly non-amenable and that the quasi-regular representation of G on ℓ2(G/H) has a uniform spectral gap. Both results are a consequence of a uniform quantitative form of ping-pong for affine transformations, which we establish here.

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References

  1. N. Alon, A. Lubotzky, A. Wigderson, Semi-direct product in groups and zig-zag product in graphs: connections and applications (extended abstract), in: 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, 2001, pp. 630–637.

  2. G. N. Arzhantseva, J. Burillo, M. Lustig, L. Reeves, H. Short, E. Ventura, Uniform non-amenability, Adv. Math. 197 (2005), no. 2, 499–522.

    Article  MathSciNet  Google Scholar 

  3. L. Auslander, The structure of complete locally affine manifolds, Topology 3 (1964), 131–139.

    Article  MathSciNet  Google Scholar 

  4. B. Bekka, P. de la Harpe, A. Valette, Kazhdan's Property (T), New Mathematical Monographs, Vol. 11, Cambridge University Press, Cambridge, 2008.

  5. M. B. Bekka, R. Curtis, On Mackey's irreducibility criterion for induced representations, Int. Math. Res. Not. 2003 (2003), no. 38, 2095–2101.

    Article  MathSciNet  Google Scholar 

  6. J. Bochi, Inequalities for numerical invariants of sets of matrices, Linear Algebra Appl. 368 (2003), 71–81.

    Article  MathSciNet  Google Scholar 

  7. E. Breuillard, A height gap theorem for finite subsets of \( {\mathrm{GL}}_d\left(\overline{\mathrm{\mathbb{Q}}}\right) \) and nonamenable subgroups, Ann. of Math. (2) 174 (2011), no. 2, 1057–1110.

  8. E. Breuillard, Heights on SL2 and free subgroups, in: Geometry, Rigidity, and Group Actions, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, pp. 455–493.

  9. E. Breuillard, A. Gamburd, Strong uniform expansion in SL(2, p), Geom. Funct. Anal. 20 (2010), no. 5, 1201–1209.

    Article  MathSciNet  Google Scholar 

  10. E. Breuillard, T. Gelander, On dense free subgroups of Lie groups, J. Algebra 261 (2003), no. 2, 448–467.

    Article  MathSciNet  Google Scholar 

  11. E. Breuillard, T. Gelander, Uniform independence in linear groups, Invent. Math. 173 (2008), no. 2, 225–263.

    Article  MathSciNet  Google Scholar 

  12. E. Breuillard, B. Green, R. Guralnick, T. Tao, Expansion in finite simple groups of Lie type, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 6, 1367–1434.

    Article  MathSciNet  Google Scholar 

  13. E. Breuillard, B. Green, T. Tao, Approximate subgroups of linear groups, Geom. Funct. Anal. 21 (2011), no. 4, 774–819.

    Article  MathSciNet  Google Scholar 

  14. M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math. 413 (1991), 36–67.

    MathSciNet  MATH  Google Scholar 

  15. S. G. Dani M. Keane, Ergodic invariant measures for actions of SL(2, Z), Ann. Inst. H. Poincaré Sect. B (N.S.) 15 (1979), no. 1, 79–84.

  16. T. J. Dekker, Decompositions of sets and spaces. I, II, Indag. Math. 18 (1956), 581–589, 590–595.

  17. T. J. Dekker, Decompositions of sets and spaces. III, Indag. Math. 19 (1957), 104–107.

  18. T. J. Dekker, Paradoxical Decompositions of Sets and Spaces, Dissertation, Drukkerij Wed. G. van Soest, Amsterdam, 1958.

  19. A. Eskin, S. Mozes, H. Oh, On uniform exponential growth for linear groups, Invent. Math. 160 (2005), no. 1, 1–30.

    Article  MathSciNet  Google Scholar 

  20. G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2016.

  21. T. Gelander, A. Żuk, Dependence of Kazhdan constants on generating subsets, Israel J. Math. 129 (2002), 93–98.

    Article  MathSciNet  Google Scholar 

  22. C. Herz, Sur le phénomène de KunzeStein, C. R. Acad. Sci. Paris Sér. A–B 271 (1970), A491–A493.

  23. J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York, 1975.

  24. E. Kaniuth, K. F. Taylor, Induced Representations of Locally Compact Groups, Cambridge Tracts in Mathematics, Vol. 197, Cambridge University Press, Cambridge, 2013.

  25. Y. Katznelson, B. Weiss, The construction of quasi-invariant measures, Israel J. Math. 12 (1972), 1–4.

    Article  MathSciNet  Google Scholar 

  26. Д. А. Каждан, О связи дуалъного пространства группы со строениует ее замкнутых подгрупп, Функц. анализ и его прил. 1 (1967), вып. 1, 71–74. Engl. transl.: D. A. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Functional. Anal. and its Appl. 1 (1967), 63–67.

  27. E. Lindenstrauss, P. P. Varjú, Spectral gap in the group of affine transformations over prime fields, Ann. Fac. Sci. Toulouse Math. (6) 25 (2016), no. 5, 969–993.

  28. A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, with an appendix by J. D. Rogawski, Progress in Mathematics, Vol. 125, Birkhäuser Verlag, Basel, 1994.

  29. A. Lubotzky, B. Weiss, Groups and expanders, in: Expanding Graphs (Princeton, NJ, 1992), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 10, Amer. Math. Soc., Providence, RI, 1993, pp. 95–109.

  30. Г. А. Маргулис, Явные конструкции расширителей, Пробп. передачи информ. 9 (1973), вып. 9, 71–80. Engl. transl.: G. A. Margulis, Explicit constructions of concentrators, Problems Inform. Transmission 9 (1973), no. 4, 325–332.

  31. Г. А. Маргулис, Свободные вполне разрывные группы аффинных преобразований, Докл. АН СССР 272 (1983), вып. 4, 785–788. Engl. transl.: G. A. Margulis, Free completely discontinuous groups of affine transformations, Sov. Math., Dokl. 28 (1983), 435–439.

  32. Г. А. Маргулис, Полные аффинные локально плоские многообразиях со свободной фундаментальной группой, Зап. научн. сем. ЛОМИ 134 (1984), 190–205. Engl. transl.: G. A. Margulis, Complete affine locally at manifolds with free fundamental group, J. Sov. Math. 36 (1987), 129–139.

  33. J. S. Milne, Algebraic Groups: The Theory of Group schemes of Finite Type over a Field, Cambridge Studies in Advanced Mathematics, Vol. 170, Cambridge University Press, Cambridge, 2017.

  34. J. Milnor, On fundamental groups of complete affinely at manifolds, Adv. in Math. 25 (1977), no. 2, 178–187.

    Article  Google Scholar 

  35. M. V. Nori, On subgroups of GLn(Fp), Invent. Math. 88 (1987), no. 2, 257–275.

    Article  MathSciNet  Google Scholar 

  36. D. Osin, D. Sonkin, Uniform Kazhdan groups, arXiv:math/0606012 (2006).

  37. D. V. Osin, Weakly amenable groups, in: Computational and Statistical Group Theory (Las Vegas, NV/Hoboken, NJ, 2001), Contemp. Math., Vol. 298, Amer. Math. Soc., Providence, RI, 2002, pp. 105–113.

  38. M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York, 1972.

  39. K. Satô, A locally commutative free group acting on the plane, Fund. Math. 180 (2003), no. 1, 25–34.

  40. Y. Shalom, Bounded generation and Kazhdan’s property (T), Inst. Hautes Études Sci. Publ. Math. 90 (1999), 145–168.

  41. Y. Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 3, 833–863.

    Article  MathSciNet  Google Scholar 

  42. A. Tarski, Sur les fonctions additives dans les classes abstraites et leur application au problème de la mesure, C. R. Soc. Sci. Varsovie, Cl. III 22 (1929), 114–117.

  43. A. Tarski, Algebraische Fassung des Maßproblems, Fundam. Math. 31 (1938), 47–66.

    Article  Google Scholar 

  44. S. Thomas, A descriptive view of unitary group representations, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 7, 1761–1787.

    Article  MathSciNet  Google Scholar 

  45. J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.

    Article  MathSciNet  Google Scholar 

  46. G. Tomkowicz, S. Wagon, The BanachTarski Paradox, Encyclopedia of Mathematics and its Applications, Vol. 163, 2nd edition, Cambridge University Press, New York, 2016.

  47. V. S. Varadarajan, Geometry of Quantum Theory, 2nd edition, Springer-Verlag, New York, 1985.

  48. S. Wagon, The BanachTarski Paradox, Cambridge University Press, Cambridge, 1993.

    MATH  Google Scholar 

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PHAM, L.L. UNIFORM KAZHDAN CONSTANTS AND PARADOXES OF THE AFFINE PLANE. Transformation Groups 27, 239–269 (2022). https://doi.org/10.1007/s00031-020-09600-5

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