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Non \(p\)-norm approximated groups. (English) Zbl 1500.22008

This paper is a continuation of a joint paper of the first author [M. De Chiffre et al., Forum Math. Sigma 8, Paper No. e18, 37 p. (2020; Zbl 1456.22002)], where it was shown that there exists a finitely presented group which cannot be approximated by almost-homomorphisms to the unitary groups \(U(n)\) with the Hilbert-Schmidt norm. Here the authors show that there is a finitely presented group which cannot be approximated by almost-homomorphisms to the unitary groups \(U(n)\) in the Schatten-\(p\)-norm, for any \(1<p<\infty\), answering a question asked by Andreas Thom in his ICM18 lecture.The proof is similar to the one used in the above cited article for the case \(p = 2\), plus some further cohomology vanishing results.

MSC:

22F10 Measurable group actions
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
20E99 Structure and classification of infinite or finite groups
46B08 Ultraproduct techniques in Banach space theory

Citations:

Zbl 1456.22002
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References:

[1] Arzhantseva, G.; Păunescu, L., Almost commuting permutations are near commuting permutations, J. Funct. Anal., 269, 745-757 (2015) · Zbl 1368.20025
[2] Bader, U.; Gelander, T.; Monod, N., A fixed point theorem for L^1spaces, Invent. Math., 189, 143-148 (2012) · Zbl 1247.46007
[3] Ballmann, W.; Światkowski, J., On L^2-cohomology and property (T) for automorphism groups of polyhedral cell complexes, Geom. Funct. Anal., 7, 615-645 (1997) · Zbl 0897.22007
[4] Bergh, J.; Löfström, J., Interpolation Spaces. An Introduction (1976), Berlin-New York: Springer-Verlag, Berlin-New York · Zbl 0344.46071
[5] Blackadar, B.; Kirchberg, E., Generalized inductive limits of finite-dimensional C*-algebras, Math. Ann., 307, 343-380 (1997) · Zbl 0874.46036
[6] Borel, A.; Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups (2000), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0980.22015
[7] Connes, A., Classification of injective factors. Cases II_1, II_∞, III_λ, λ ≠ 1, Ann. of Math. (2), 104, 73-115 (1976) · Zbl 0343.46042
[8] M. De Chiffre, L. Glebsky, A. Lubotzky and A. Thom, Stability, cohomology vanishing, and non-approximable groups, Forum of Math. Sigma, 2020, doi:10.1017/fms.2020.5 · Zbl 1456.22002
[9] Deligne, P., Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Acad. Sci. Paris Sér. A-B, 287, A203-A208 (1978) · Zbl 0416.20042
[10] Dymara, J.; Januszkiewicz, T., Cohomology of buildings and their automorphism groups, Invent. Math., 150, 579-627 (2002) · Zbl 1140.20308
[11] Gaal, S. A., Linear Analysis and Representation Theory (1973), New York-Heidelberg: Springer-Verlag, New York-Heidelberg · Zbl 0275.43008
[12] Garland, H., p-adic curvature and the cohomology of discrete subgroups of p-adic groups, Ann. of Math. (2), 97, 375-423 (1973) · Zbl 0262.22010
[13] Glebsky, L.; Manuel Rivera, L., Sofic groups and profinite topology on free groups, J. Algebra, 320, 3512-3518 (2008) · Zbl 1162.20015
[14] Lyndon, R. C.; Schupp, P. E., Combinatorial Group Theory (2001), Berlin: Springer-Verlag, Berlin · Zbl 0997.20037
[15] Megginson, R. E., An Introduction to Banach Space Theory (1998), New York: Springer-Verlag, New York · Zbl 0910.46008
[16] Megrelishvili, M. G., Fragmentability and continuity of semigroup actions, Semigroup Forum, 57, 101-126 (1998) · Zbl 0916.47029
[17] Oppenheim, I., Averaged projections, angles between groups and strengthening of Banach property (T), Math. Ann., 367, 623-666 (2017) · Zbl 1475.22014
[18] Oppenheim, I., Vanishing of cohomology with coefficients in representations on Banach spaces of groups acting on buildings, Comment. Math. Helv., 92, 389-428 (2017) · Zbl 1475.20088
[19] Pestov, V. G., Hyperlinear and sofic groups: a brief guide, Bull. Symbolic Logic, 14, 449-480 (2008) · Zbl 1206.20048
[20] Pisier, G., Some applications of the complex interpolation method to Banach lattices, J. Analyse Math., 35, 264-281 (1979) · Zbl 0427.46048
[21] Pisier, G.; Xu, Q., Non-commutative L^p-spaces, Handbook of the Geometry of Banach Spaces, Vol. 2, 1459-1517 (2003), Amsterdam: North-Holland, Amsterdam · Zbl 1046.46048
[22] de la Salle, M., Towards strong Banach property (T) for SL(3, ℝ), Israel J. Math., 211, 105-145 (2016) · Zbl 1362.46028
[23] Sundaresan, K., The Radon-Nikodým theorem for Lebesgue-Bochner function spaces, J. Functional Analysis, 24, 276-279 (1977) · Zbl 0341.46019
[24] Thom, A., Finitary approximations of groups and their applications, Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018. Vol. III. Invited lectures, 1779-1799 (2018), Hackensack, NJ: World Scientific, Hackensack, NJ · Zbl 1445.20039
[25] Yost, D., Asplund spaces for beginners, Acta Univ. Carolin. Math. Phys., 34, 159-177 (1993) · Zbl 0815.46022
[26] Zimmer, R. J., Ergodic Theory and Semisimple Groups (1984), Basel: Birkhäuser Verlag, Basel · Zbl 0571.58015
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