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.
Similar content being viewed by others
References
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.
G. N. Arzhantseva, J. Burillo, M. Lustig, L. Reeves, H. Short, E. Ventura, Uniform non-amenability, Adv. Math. 197 (2005), no. 2, 499–522.
L. Auslander, The structure of complete locally affine manifolds, Topology 3 (1964), 131–139.
B. Bekka, P. de la Harpe, A. Valette, Kazhdan's Property (T), New Mathematical Monographs, Vol. 11, Cambridge University Press, Cambridge, 2008.
M. B. Bekka, R. Curtis, On Mackey's irreducibility criterion for induced representations, Int. Math. Res. Not. 2003 (2003), no. 38, 2095–2101.
J. Bochi, Inequalities for numerical invariants of sets of matrices, Linear Algebra Appl. 368 (2003), 71–81.
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.
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.
E. Breuillard, A. Gamburd, Strong uniform expansion in SL(2, p), Geom. Funct. Anal. 20 (2010), no. 5, 1201–1209.
E. Breuillard, T. Gelander, On dense free subgroups of Lie groups, J. Algebra 261 (2003), no. 2, 448–467.
E. Breuillard, T. Gelander, Uniform independence in linear groups, Invent. Math. 173 (2008), no. 2, 225–263.
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.
E. Breuillard, B. Green, T. Tao, Approximate subgroups of linear groups, Geom. Funct. Anal. 21 (2011), no. 4, 774–819.
M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math. 413 (1991), 36–67.
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.
T. J. Dekker, Decompositions of sets and spaces. I, II, Indag. Math. 18 (1956), 581–589, 590–595.
T. J. Dekker, Decompositions of sets and spaces. III, Indag. Math. 19 (1957), 104–107.
T. J. Dekker, Paradoxical Decompositions of Sets and Spaces, Dissertation, Drukkerij Wed. G. van Soest, Amsterdam, 1958.
A. Eskin, S. Mozes, H. Oh, On uniform exponential growth for linear groups, Invent. Math. 160 (2005), no. 1, 1–30.
G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2016.
T. Gelander, A. Żuk, Dependence of Kazhdan constants on generating subsets, Israel J. Math. 129 (2002), 93–98.
C. Herz, Sur le phénomène de Kunze–Stein, C. R. Acad. Sci. Paris Sér. A–B 271 (1970), A491–A493.
J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York, 1975.
E. Kaniuth, K. F. Taylor, Induced Representations of Locally Compact Groups, Cambridge Tracts in Mathematics, Vol. 197, Cambridge University Press, Cambridge, 2013.
Y. Katznelson, B. Weiss, The construction of quasi-invariant measures, Israel J. Math. 12 (1972), 1–4.
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.
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.
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.
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.
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.
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.
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.
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.
J. Milnor, On fundamental groups of complete affinely at manifolds, Adv. in Math. 25 (1977), no. 2, 178–187.
M. V. Nori, On subgroups of GLn(Fp), Invent. Math. 88 (1987), no. 2, 257–275.
D. Osin, D. Sonkin, Uniform Kazhdan groups, arXiv:math/0606012 (2006).
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.
M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York, 1972.
K. Satô, A locally commutative free group acting on the plane, Fund. Math. 180 (2003), no. 1, 25–34.
Y. Shalom, Bounded generation and Kazhdan’s property (T), Inst. Hautes Études Sci. Publ. Math. 90 (1999), 145–168.
Y. Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 3, 833–863.
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.
A. Tarski, Algebraische Fassung des Maßproblems, Fundam. Math. 31 (1938), 47–66.
S. Thomas, A descriptive view of unitary group representations, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 7, 1761–1787.
J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.
G. Tomkowicz, S. Wagon, The Banach–Tarski Paradox, Encyclopedia of Mathematics and its Applications, Vol. 163, 2nd edition, Cambridge University Press, New York, 2016.
V. S. Varadarajan, Geometry of Quantum Theory, 2nd edition, Springer-Verlag, New York, 1985.
S. Wagon, The Banach–Tarski Paradox, Cambridge University Press, Cambridge, 1993.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-020-09600-5