## Duke Mathematical Journal

### Approximate lattices

#### Abstract

In this article we introduce and study uniform and nonuniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasicrystals (Meyer sets) in lcsc Abelian groups. We show that envelopes of strong approximate lattices are unimodular and that approximate lattices in nilpotent groups are uniform. We also establish several results relating properties of approximate lattices and their envelopes. For example, we prove a version of the Milnor–Schwarz lemma for uniform approximate lattices in compactly generated lcsc groups, which we then use to relate the metric amenability of uniform approximate lattices to the amenability of the envelope. Finally we extend a theorem of Kleiner and Leeb to show that the isometry groups of irreducible higher-rank symmetric spaces of noncompact type are quasi-isometrically rigid with respect to finitely generated approximate groups.

#### Article information

Source
Duke Math. J., Volume 167, Number 15 (2018), 2903-2964.

Dates
Revised: 21 May 2018
First available in Project Euclid: 3 October 2018

https://projecteuclid.org/euclid.dmj/1538532050

Digital Object Identifier
doi:10.1215/00127094-2018-0028

Mathematical Reviews number (MathSciNet)
MR3865655

Zentralblatt MATH identifier
06982210

#### Citation

Björklund, Michael; Hartnick, Tobias. Approximate lattices. Duke Math. J. 167 (2018), no. 15, 2903--2964. doi:10.1215/00127094-2018-0028. https://projecteuclid.org/euclid.dmj/1538532050

#### References

• [1] M. Abért, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, and I. Samet, On the growth of $L^{2}$-invariants for sequences of lattices in Lie groups, Ann. of Math. (2) 185 (2017), 711–790.
• [2] M. Abért, Y. Glasner, and B. Virág, Kesten’s theorem for invariant random subgroups, Duke Math. J. 163 (2014), 465–488.
• [3] M. Baake and U. Grimm, Aperiodic Order, Vol. 1, Encyclopedia Math. Appl. 149, Cambridge Univ. Press, Cambridge, 2013.
• [4] U. Bader, A. Furman, and R. Sauer, On the structure and arithmeticity of lattice envelopes, C. R. Math. Acad. Sci. Paris 353 (2015), 409–413.
• [5] M. Björklund, Product set phenomena for measured groups, Ergodic Theory Dynam. Systems, published electronically 4 May 2017.
• [6] M. Björklund and A. Fish, Approximate invariance for ergodic actions of amenable groups, preprint, arXiv:1607.02575v3 [math.DS].
• [7] M. Björklund, T. Hartnick, and F. Pogorzelski, Aperiodic order and spherical diffraction, I: Auto-correlation of regular model sets, Proc. Lond. Math. Soc. (3) 116 (2018), 957–996.
• [8] A. Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179–188.
• [9] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535.
• [10] J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of $\operatorname{SL}_{2}(\mathbb{F}_{p})$, Ann. of Math. (2) 167 (2008), 625–642.
• [11] E. Breuillard, “A brief introduction to approximate groups” in Thin Groups and Superstrong Approximation, Math. Sci. Res. Inst. Publ. 61, Cambridge Univ. Press, Cambridge, 2014, 23–50.
• [12] E. Breuillard, Lectures on approximate groups, lecture notes, https://www.math.u-psud.fr/~breuilla/ClermontLectures.pdf.
• [13] E. Breuillard, B. Green, and T. Tao, The structure of approximate groups, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115–221.
• [14] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss. 319, Springer, Berlin, 1999.
• [15] P.-E. Caprace and N. Monod, Isometry groups of non-positively curved spaces: discrete subgroups, J. Topol. 2 (2009), 701–746.
• [16] Y. Cornulier and P. de la Harpe, Metric Geometry of Locally Compact Groups, EMS Tracts in Math. 25, Eur. Math. Soc. (EMS), Zürich, 2016.
• [17] L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications, I, Cambridge Stud. Adv. Math. 18, Cambridge Univ. Press, Cambridge, 1990.
• [18] T. Dymarz, Envelopes of certain solvable groups, Comment. Math. Helv. 90 (2015), 195–224.
• [19] G. Elek and G. Tardos, On roughly transitive amenable graphs and harmonic Dirichlet functions, Proc. Amer. Math. Soc. 128 (2000), 2479–2485.
• [20] P. Erdős and E. Szemerédi, “On sums and products of integers” in Studies in Pure Mathematics, Birkhäuser, Basel, 1983, 213–218.
• [21] A. Eskin and B. Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997), 653–692.
• [22] E. Følner, Note on a generalization of a theorem of Bogolioùboff, Math. Scand. 2 (1954), 224–226.
• [23] G. A. Freĭman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr. 37, Amer. Math. Soc., Providence, 1973.
• [24] A. Furman, Mostow-Margulis rigidity with locally compact targets, Geom. Funct. Anal. 11 (2001), 30–59.
• [25] E. Glasner and B. Weiss, “Uniformly recurrent subgroups” in Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math. 631, Amer. Math. Soc., Providence, 2015, 63–75.
• [26] M.-P. Gong, Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and $\mathbb{R}$), Ph.D. dissertation, University of Waterloo, Waterloo, Canada, 1998.
• [27] M. Gromov, “Asymptotic invariants of infinite groups,” in Geometric Group Theory, Volume 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press, Cambridge, 1993, 1–295.
• [28] H. A. Helfgott, Growth and generation in $\operatorname{SL}_{2}(\mathbb{Z}/p\mathbb{Z})$, Ann. of Math. (2) 167 (2008), 601–623.
• [29] V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), 457–490.
• [30] B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 115–197.
• [31] Y. Meyer, Algebraic Numbers and Harmonic Analysis, North-Holland Math. Libr. 2, Elsevier, New York, 1972.
• [32] R. V. Moody, “Meyer sets and their duals” in The Mathematics of Long-Range Aperiodic Order (Waterloo, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 489, Kluwer, Dordrecht, 1997, 403–441.
• [33] L. Mosher, M. Sageev, and K. Whyte, Quasi-actions on trees, I: Bounded valence, Ann. of Math. (2) 158 (2003), 115–164.
• [34] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1–60.
• [35] F. Paulin, “De la geometrie et de la dynamique de $\operatorname{SL}_{n}(\mathbb{R})$ et $\operatorname{SL}_{n}(\mathbb{Z})$” in Sur la dynamique des groupes de matrices et applications arithmétiques, Ed. Éc. Polytech., Palaiseau, 2007, 47–110.
• [36] H. Plünnecke, Eigenschaften und Abschätzungen von Wirkungsfunktionen, Gesellschaft für Mathematik und Datenverarbeitung, Bonn, 1969.
• [37] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb. (3) 68, Springer, New York, 1972.
• [38] A. Raugi, A general Choquet-Deny theorem for nilpotent groups, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), 677–683.
• [39] J. Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann. 257 (1981), 31–42.
• [40] I. Z. Ruzsa, “An analog of Freiman’s theorem in groups” in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Paris, 1999, 323–326.
• [41] J. Scheunemann, Two-step nilpotent Lie algebras, J. Algebra 7 (1967), 152–159.
• [42] T. Tao, Product set estimates for non-commutative groups, Combinatorica 28 (2008), 547–594.