×

zbMATH — the first resource for mathematics

A height gap theorem for finite subsets of \(\mathrm{GL}_{d}(\overline{\mathbb{Q}})\) and nonamenable subgroups. (English) Zbl 1243.11071
The author introduces a conjugation invariant height \(\hat{h}(F)\) on finite subsets of matrices \(F\) in \(\mathrm{GL}_{d}(\overline{\mathbb Q})\), where \(\overline{\mathbb{Q}}\) is the field of algebraic numbers, which is well suited to the study of geometric and arithmetic behavior of power sets \(F^{n} = F\cdot\cdot\cdot F\) for \(n\in\bar{\mathbb Q}\). Among many interesting results of the paper the following two theorems are outstanding: Let \(F\) be a finite subset of \(\mathrm{GL}_{d}(\overline{\mathbb Q})\) generating a nonamenable subgroup that acts strongly irreducibly. Then there exists a constant \(\varepsilon(d)>0\) such that \(\hat{h}(F)> \varepsilon(d)\). If \(F\) is a finite subset of \(\mathrm{GL}_{d}(\overline{\mathbb Q})\) generating a nonvirtually solvable subgroup, then there exists a constant \(\varepsilon(d)>0\) such that \(\hat{h}(F)> \varepsilon(d)\). Using these two theorems the author can for instance prove the following statements:
There is an \(n(d)\in\mathbb N\) such that if \(K\) is a field and \(F\) is a finite symmetric subset of \(\mathrm{GL}_{d}(K)\) containing \(1\) which generates a nonvirtually solvable subgroup, then \(F^n\) contains two elements \(A\) and \(B\) which generate a non-abelian free subgroup.
There is an an integer \(n(d)\in\mathbb N\) such that if \(K\) is a field and \(F\) is finite subset of \(\mathrm{GL}_{d}(K)\) which generates an infinite subgroup, then \((F\cup F^{-1})^n\) contains an element of infinite order.
The interpretation of \(\hat{h}(F)\) in terms of spectral radius allows the author to derive the following two statements:
There are constants \(n(d)\in\mathbb N\) and \(c(d)\in\mathbb N\) such that if \(F\) is any finite subset of \(\mathrm{GL}_{d}((\overline{\mathbb Q})\) containing \(1\), then there is some \(A\in F^n\) and some eigenvalue \(\lambda\) of \(A\) such that \[ h(\lambda)\geq \frac{1}{|F^{c}|}\cdot \hat{h}(F), \] where \(h(\lambda)\) is the height of \(\lambda\) in sense of [E. Bompieri and W. Gubler, Heights in Diophantic Geometry. New Mathematical Monographs 4. Cambridge: Cambridge University Press. (2006; Zbl 1115.11034)].
There are constants \(n(d)\in\mathbb N\) and \(\varepsilon(d)> 0\) such that if \(F\) is any finite subset of \(\mathrm{GL}_{d}(\overline{\mathbb Q})\) containing \(1\) and generating a nonvirtually solvable subgroup, then one finds a matrix \(A\in F^n\) and an eigenvalue \(\lambda\) of \(A\) such that \(h(\lambda)> \varepsilon(d)\).
If \(F\) is is a finite subset of \(\mathrm{GL}_{d}(\overline{\mathbb Q})\), then \(\hat{h}(F) = 0\) if and only if the group generated by \(F\) is virtually nilpotent.
The author introduces for \(F\) also the minimal height and compares the minimal height \(h(F)\) with the normalized height \(\hat{h}(F)\). Using estimates relating the minimal norm of \(F\) defined by the author and the matrix coefficients of the elements of \(F\) in the adjoint representation he is able to obtain a global upper bound on the height of the matrix coefficients of the finite set \(F\) of matrices in \(\mathrm{GL}_{d}(\overline{\mathbb Q})\). The author gives in the paper many hints at the relations of his results to classical statements (e.g., Margulis lemma, uniform Tits alternative, uniform version of the Burnside-Schur theorem) and to classical conjectures (e. g., the Lehmer problem).

MSC:
11G50 Heights
11R04 Algebraic numbers; rings of algebraic integers
22E40 Discrete subgroups of Lie groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. Amoroso and S. David, ”Le problème de Lehmer en dimension supérieure,” J. Reine Angew. Math., vol. 513, pp. 145-179, 1999. · Zbl 1011.11045 · doi:10.1515/crll.1999.058
[2] F. Amoroso and R. Dvornicich, ”A lower bound for the height in abelian extensions,” J. Number Theory, vol. 80, iss. 2, pp. 260-272, 2000. · Zbl 0973.11092 · doi:10.1006/jnth.1999.2451
[3] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Reading, MA: Addison-Wesley Publishing Co., 1969. · Zbl 0175.03601
[4] M. H. Baker and R. Rumely, ”Equidistribution of small points, rational dynamics, and potential theory,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 56, iss. 3, pp. 625-688, 2006. · Zbl 1234.11082 · doi:10.5802/aif.2196 · numdam:AIF_2006__56_3_625_0 · eudml:10160 · arxiv:math/0407426
[5] L. Bartholdi and Y. de Cornulier, ”Infinite groups with large balls of torsion elements and small entropy,” Arch. Math. \((\)Basel\()\), vol. 87, iss. 2, pp. 104-112, 2006. · Zbl 1136.20035 · doi:10.1007/s00013-005-1684-4 · arxiv:math/0510141
[6] Y. Bilu, ”Limit distribution of small points on algebraic tori,” Duke Math. J., vol. 89, iss. 3, pp. 465-476, 1997. · Zbl 0918.11035 · doi:10.1215/S0012-7094-97-08921-3
[7] E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Cambridge: Cambridge Univ. Press, 2006, vol. 4. · Zbl 1115.11034 · doi:10.2277/0511138091
[8] A. Borel, Linear Algebraic Groups, New York: W. A. Benjamin, 1969. · Zbl 0186.33201
[9] N. Bourbaki, Groupes et Algèbres de Lie, Hermann. · Zbl 0483.22001
[10] E. Breuillard, ”On uniform exponential growth for solvable groups,” Pure Appl. Math. Q., vol. 3, iss. 4, part 1, pp. 949-967, 2007. · Zbl 1147.20027 · doi:10.4310/PAMQ.2007.v3.n4.a4 · arxiv:math/0602076
[11] E. Breuillard and T. Gelander, ”Uniform independence in linear groups,” Invent. Math., vol. 173, iss. 2, pp. 225-263, 2008. · Zbl 1148.20029 · doi:10.1007/s00222-007-0101-y · arxiv:math/0611829
[12] E. Breuillard, ”Heights on \({ {SL}}_2\) and free subgroups,” in Geometry, Rigidity and Group Actions, Zimmer’s Festschrift, Farb, B. and Fisher, D., Eds., Chicago, IL: Chicago Univ. Press, 2011. · Zbl 1261.20047
[13] E. Breuillard, A strong Tits alternative, 2008. · Zbl 1149.20039
[14] E. Breuillard, Effective estimates for the spectral radius of a finite set of matrices. · Zbl 1243.11071
[15] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, New York: Springer-Verlag, 1999, vol. 319. · Zbl 0988.53001
[16] F. Bruhat and J. Tits, ”Groupes réductifs sur un corps local,” Inst. Hautes Études Sci. Publ. Math., iss. 41, pp. 5-251, 1972. · Zbl 0254.14017 · doi:10.1007/BF02715544 · numdam:PMIHES_1972__41__5_0 · eudml:103918
[17] P. E. Caprace.
[18] A. Chambert-Loir, ”Mesures et équidistribution sur les espaces de Berkovich,” J. Reine Angew. Math., vol. 595, pp. 215-235, 2006. · Zbl 1112.14022 · doi:10.1515/CRELLE.2006.049 · arxiv:math/0304023
[19] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, New York: Interscience Publishers, a division of John Wiley & Sons, 1962, vol. 11. · Zbl 0131.25601
[20] P. B. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago, IL: University of Chicago Press, 1996. · Zbl 0883.53003
[21] A. Eskin, S. Mozes, and H. Oh, ”On uniform exponential growth for linear groups,” Invent. Math., vol. 160, iss. 1, pp. 1-30, 2005. · Zbl 1137.20024 · doi:10.1007/s00222-004-0378-z
[22] C. Favre and J. Rivera-Letelier, ”Équidistribution quantitative des points de petite hauteur sur la droite projective,” Math. Ann., vol. 335, iss. 2, pp. 311-361, 2006. · Zbl 1175.11029 · doi:10.1007/s00208-006-0751-x
[23] T. Gelander, ”Homotopy type and volume of locally symmetric manifolds,” Duke Math. J., vol. 124, iss. 3, pp. 459-515, 2004. · Zbl 1076.53040 · doi:10.1215/S0012-7094-04-12432-7 · arxiv:math/0111165
[24] N. Iwahori and H. Matsumoto, ”On some Bruhat decomposition and the structure of the Hecke rings of \(\mathitp\)-adic Chevalley groups,” Inst. Hautes Études Sci. Publ. Math., iss. 25, pp. 5-48, 1965. · Zbl 0228.20015 · doi:10.1007/BF02684396 · numdam:PMIHES_1965__25__5_0 · eudml:103854
[25] N. Jacobson, Lie Algebras, New York: Interscience Publishers, a division of John Wiley & Sons, 1962, vol. 10. · Zbl 0121.27504
[26] D. A. Kazdan and G. A. Margulis, ”A proof of Selberg’s hypothesis,” Mat. Sb., vol. 75 (117), pp. 163-168, 1968. · Zbl 0241.22024 · doi:10.1070/SM1968v004n01ABEH002782
[27] E. Landvogt, ”Some functorial properties of the Bruhat-Tits building,” J. Reine Angew. Math., vol. 518, pp. 213-241, 2000. · Zbl 0937.20026 · doi:10.1515/crll.2000.006
[28] S. Lang, Algebra, third ed., New York: Springer-Verlag, 2002, vol. 211. · Zbl 0984.00001
[29] S. Lang, Fundamentals of Diophantine Geometry, New York: Springer-Verlag, 1983. · Zbl 0528.14013
[30] D. W. Masser and G. Wüstholz, ”Fields of large transcendence degree generated by values of elliptic functions,” Invent. Math., vol. 72, iss. 3, pp. 407-464, 1983. · Zbl 0516.10027 · doi:10.1007/BF01398396 · eudml:143029
[31] G. D. Mostow, ”Self-adjoint groups,” Ann. of Math., vol. 62, pp. 44-55, 1955. · Zbl 0065.01404 · doi:10.2307/2007099
[32] A. L. Onishchik and È. B. Vinberg, Lie Groups and Algebraic Groups, New York: Springer-Verlag, 1990. · Zbl 0722.22004
[33] J. Pineiro, L. Szpiro, and T. J. Tucker, ”Mahler measure for dynamical systems on \({\mathbb P}^1\) and intersection theory on a singular arithmetic surface,” in Geometric Methods in Algebra and Number Theory, Boston, MA: Birkhäuser, 2005, vol. 235, pp. 219-250. · Zbl 1101.11020 · doi:10.1007/0-8176-4417-2_10
[34] M. S. Raghunathan, Discrete Subgroups of Lie Groups, New York: Springer-Verlag, 1972, vol. 68. · Zbl 0254.22005
[35] A. Schinzel, Polynomials with Special Regard to Reducibility, Cambridge: Cambridge Univ. Press, 2000, vol. 77. · Zbl 0956.12001 · doi:10.1017/CBO9780511542916
[36] I. Schur, Über Gruppen periodischer Substitutionen, Sitzber. Preuss. Akad. Wiss., 1911. · JFM 42.0155.01
[37] Y. Shalom, ”Explicit Kazhdan constants for representations of semisimple and arithmetic groups,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 50, iss. 3, pp. 833-863, 2000. · Zbl 0966.22004 · doi:10.5802/aif.1775 · numdam:AIF_2000__50_3_833_0 · eudml:75440
[38] C. Smyth, ”The Mahler measure of algebraic numbers: a survey,” in Number Theory and Polynomials, Cambridge: Cambridge Univ. Press, 2008, vol. 352, pp. 322-349. · Zbl 1334.11081 · doi:10.1017/CBO9780511721274.021 · arxiv:math/0701397
[39] L. Szpiro, E. Ullmo, and S. Zhang, ”Équirépartition des petits points,” Invent. Math., vol. 127, iss. 2, pp. 337-347, 1997. · Zbl 0991.11035 · doi:10.1007/s002220050123
[40] R. Steinberg, Lectures on Chevalley Groups, New Haven, Conn.: Yale University, 1968. · Zbl 1196.22001
[41] V. Talamanca, ”A Gelfand-Beurling type formula for heights on endomorphism rings,” J. Number Theory, vol. 83, iss. 1, pp. 91-105, 2000. · Zbl 0965.16018 · doi:10.1006/jnth.1999.2506 · arxiv:math/9912251
[42] J. Tits, ”Free subgroups in linear groups,” J. Algebra, vol. 20, pp. 250-270, 1972. · Zbl 0236.20032 · doi:10.1016/0021-8693(72)90058-0
[43] W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton, NJ: Princeton Univ. Press, 1997, vol. 35. · Zbl 0873.57001
[44] E. Ullmo, ”Positivité et discrétion des points algébriques des courbes,” Ann. of Math., vol. 147, iss. 1, pp. 167-179, 1998. · Zbl 0934.14013 · doi:10.2307/120987
[45] H. C. Wang, ”Topics on totally discontinuous groups,” in Symmetric Spaces, New York: Dekker, 1972, vol. 8, pp. 459-487. · Zbl 0232.22018
[46] B. A. F. Wehrfritz, Infinite Linear Groups. An Account of the Group-Theoretic Properties of Infinite Groups of Matrices, New York: Springer-Verlag, 1973, vol. 76. · Zbl 0261.20038
[47] S. Zhang, ”Small points and adelic metrics,” J. Algebraic Geom., vol. 4, iss. 2, pp. 281-300, 1995. · Zbl 0861.14019
[48] S. Zhang, ”Equidistribution of small points on abelian varieties,” Ann. of Math., vol. 147, iss. 1, pp. 159-165, 1998. · Zbl 0991.11034 · doi:10.2307/120986
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.