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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
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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
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