Skip to main content
SpringerLink
Log in

New Dimensional Estimates for Subvarieties of Linear Algebraic Groups

  • Original Article
  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

For every connected, almost simple linear algebraic group \(G\le \textrm{GL}_{n}\) over a large enough field K, every subvariety \(V\subseteq G\), and every finite generating set \(A\subseteq G(K)\), we prove a general dimensional bound, that is, a bound of the form

$$ |A\cap V(\overline{K})|\le C_{1}|A^{C_{2}}|^{\frac{\dim (V)}{\dim (G)}} $$

with \(C_{1}\), \(C_{2}\) depending only on n, \(\textrm{deg}(V)\). The dependence of \(C_1\) on n (or rather on \(\dim (V)\)) is doubly exponential, whereas \(C_2\) (which is independent of \(\textrm{deg}(V)\)) depends simply exponentially on n. Bounds of this form have proved useful in the study of growth in linear algebraic groups since 2005 (Helfgott) and, before then, in the study of subgroup structure (Larsen–Pink: A a subgroup). In bounds for general V and G available before our work, the dependence of \(C_1\) and \(C_2\) on n was of exponential-tower type. We draw immediate consequences regarding diameter bounds for untwisted classical groups \(G(\mathbb {F}_{q})\). (In a separate paper, we derive stronger diameter bounds from stronger dimensional bounds we prove for specific families of varieties V.)

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. First available in manuscript form in 1998.

  2. Many sources, such as [18] and [37], reserve the word variety for what we call irreducible variety, and call closed algebraic sets what we call varieties. We follow instead the same convention that [42] and [11] follow.

  3. This is a classical meaning of “generic”, used in the literature on growth in groups (e.g., [20, Section 2.5.2] and [4, Definition 3.6]) but also elsewhere. The meaning of “generic” in scheme theory is surely inspired by it but not identical: the classical meaning is still current in conversation, but “generic point” in scheme theory means something else – a non-closed point whose closure contains all closed points of a variety, i.e., all its points in the classical sense (see [37, Section II.1, Definition 1] and [18, Section II.2, Example 2.3.3]). Cf. the discussion at the beginning of [32, Section 2].

  4. Ideal-theoretically: \(g_1\) and \(g_2\) are equivalent modulo I(X). We are defining morphisms defined on X as restrictions of morphisms \(\mathbb {A}^m\rightarrow \mathbb {A}^{m'}\) so that there is no ambiguity in speaking of the maximum degree \(\textrm{mdeg}(f)\) (Section 2.1.2).

  5. Kindly shown to us by Miles Reid.

  6. Bound (2.3) was given by user Angelo on MathOverflow in 2011 in reply to question 63451 by HH.

  7. Some authors call these groups simple. “Such an algebraic group is called simple (or almost simple, if we wish to emphasize that the group need not be simple as an abstract group)” ([27, p. 168]).

  8. This can be shown easily by induction on m. Let \(V\subseteq \mathbb {A}^m\) contain \(\mathbb {A}^m(K)\). Then, for every \(x_m\in K\), the intersection \(V\cap (\mathbb {A}^{m-1}\times \{x_m\})\) contains every point of \(\mathbb {A}^{m-1}(K)\times \{x_m\}\), and thus, by the inductive hypothesis, it must equal \(\mathbb {A}^{m-1}\times \{x_m\}\). However, V cannot contain the infinitely many \((m-1)\)-dimensional varieties \(\mathbb {A}^{m-1}\times \{x_n\}\) unless V is m-dimensional. Hence, \(V = \mathbb {A}^m\).

  9. We thank user Jeff Adler on MathOverflow for this reference, and for pointing out that the case \(G_{2}(\mathbb {F}_{2})\) was also inadvertently omitted (question 452013).

References

  1. Babai, L.: On the diameter of Eulerian orientations of graphs. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 822–831. ACM, New York (2006)

  2. Bajpai, J., Dona, D., Helfgott, H.A.: Growth estimates and diameter bounds for classical Chevalley groups. arXiv:2110.02942 (2021)

  3. Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Mathematics, vol. 126. Springer, New York (1991)

  4. Breuillard, E., Green, B., Tao, T.: Approximate subgroups of linear groups. Geom. Funct. Anal. 21, 774–819 (2011)

    Article  MathSciNet  Google Scholar 

  5. Cafure, A., Matera, G.: Improved explicit estimates on the number of solutions of equations over a finite field. Finite Fields Appl. 12, 155–185 (2006)

    Article  MathSciNet  Google Scholar 

  6. Carter, R.W.: Simple Groups of Lie Type, Reprinted edn. John Wiley & Sons, London (1989)

  7. Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, reprinted edn. John Wiley & Sons, Chichester (1993)

  8. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford (1985)

  9. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. With Comput. Assist. from J.G. Thackray. Clarendon Press, Oxford (1985)

  10. Cooperstein, B.N.: The fifty-six-dimensional module for \(E_{7}\): I. A four form for \(E_{7}\). J. Algebra 173, 361–389 (1995)

  11. Danilov, V.I., Shokurov, V.N.: Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes. Encyclopaedia of Mathematical Sciences, vol. 23. Springer-Verlag, Berlin (1998)

  12. Dat, J.F., Orlik, S., Rapoport, M.: Period Domains over Finite and \(p\)-Adic Fields. Cambridge Tracts in Mathematics, vol. 183. Cambridge University Press, Cambridge (2010)

  13. Dieudonné, J.: La géométrie des Groupes Classiques. Ergebnisse der Mathematik und ihrer Grenzgebiete (Neue Folge), vol. 5. Springer-Verlag, Berlin (1955)

  14. Eskin, A., Mozes, S., Oh, H.: On uniform exponential growth for linear groups. Invent. Math. 160, 1–30 (2005)

  15. Freudenthal, H.: Elementarteilertheorie der komplexen orthogonalen und symplektischen Gruppen. Indag. Math. (Proc.) 55, 199–201 (1952)

    Article  MathSciNet  Google Scholar 

  16. Fulton, W.: Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete: a series of Modern Surveys in Mathematics. Folge 3, vol. 2. Springer, Berlin (1984)

  17. Harder, G.: Lectures on Algebraic Geometry II. Basic Concepts, Coherent Cohomology, Curves and their Jacobians. Aspects Math., vol. E39. Vieweg+Teubner, Wiesbaden (2011)

  18. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer-Verlag, New York (1977)

  19. Helfgott, H.A.: Growth and generation in \(\text{SL}_{2}(\mathbb{Z}/\text{p }\mathbb{Z})\). Ann. Math. (2)167, 601–623 (2008)

  20. Helfgott, H.A.: Growth in \(\rm SL _3(\mathbb{Z} /p\mathbb{Z} )\). J. Eur. Math. Soc. 13, 761–851 (2011)

    Article  MathSciNet  Google Scholar 

  21. Helfgott, H.A.: Growth and expansion in algebraic groups over finite fields. In: Bucur, A., Zureick-Brown, D. (eds.) Analytic Methods in Arithmetic Geometry. Contemporary Mathematics, vol. 740, pp. 71–111. Amer. Math. Soc, Providence, RI (2019)

    Chapter  Google Scholar 

  22. Hogeweij, G.M.D.: Almost-Classical Lie Algebras. I, II. Nederl. Akad. Wetensch. Indag. Math. 44, 441–452, 453–460 (1982)

  23. Hrushovski, E.: Stable group theory and approximate subgroups. J. Amer. Math. Soc. 25, 189–243 (2012)

    Article  MathSciNet  Google Scholar 

  24. Hrushovski, E., Pillay, A.: Definable subgroups of algebraic groups over finite fields. J. Reine Angew. Math. 462, 69–91 (1995)

    MathSciNet  Google Scholar 

  25. Hrushovski, E., Wagner, F.: Counting and dimensions. In: Chatzidakis, Z., Macpherson, D., Pillay, A., Wilkie, A. (eds.) Model Theory with Applications to Algebra and Analysis, vol. 2, pp. 161–176. London Mathematical Society Lecture Note Series, vol. 350. Cambridge University Press, Cambridge (2008)

  26. Humphreys, J.E.: Conjugacy Classes in Semisimple Algebraic Groups. Mathematical Surveys and Monographs, vol. 43. American Mathematical Society, Providence, RI (1995)

  27. Humphreys, J.E.: Linear Algebraic Groups, 4th edn. Graduate Texts in Mathematics, vol. 21. Springer-Verlag, New York (1995)

  28. Kleidman, P., Liebeck, M.: The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, vol. 129. Cambridge University Press, Cambridge (1990)

  29. Lachaud, G., Rolland, R.: On the number of points of algebraic sets over finite fields. J. Pure Appl. Algebra 219, 5117–5136 (2015)

    Article  MathSciNet  Google Scholar 

  30. Lang, S., Weil, A.: Number of points of varieties in finite fields. Amer. J. Math. 76, 819–827 (1954)

    Article  MathSciNet  Google Scholar 

  31. Larsen, M.: Word maps have large image. Isr. J. Math. 139, 149–156 (2004)

    Article  MathSciNet  Google Scholar 

  32. Larsen, M.J., Pink, R.: Finite subgroups of algebraic groups. J. Amer. Math. Soc. 24, 1105–1158 (2011)

    Article  MathSciNet  Google Scholar 

  33. Lazarsfeld, R.: Excess intersection of divisors. Compos. Math. 43, 281–296 (1981)

    MathSciNet  Google Scholar 

  34. Lehrer, G.I.: Rational tori, semisimple orbits and the topology of hyperplane complements. Comment. Math. Helv. 67, 226–251 (1992)

    Article  MathSciNet  Google Scholar 

  35. Lemire, N., Popov, V.L., Reichstein, Z.: Cayley groups. J. Amer. Math. Soc. 19, 921–967 (2006)

    Article  MathSciNet  Google Scholar 

  36. Milne, J.S.: Algebraic Groups. Cambridge Studies in Advanced Mathematics, vol. 170. Cambridge University Press, Cambridge (2017)

  37. Mumford, D.: The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol. 1358, expanded edn. Springer-Verlag, Berlin (1999)

  38. Nikolov, N., Pyber, L.: Product decompositions of quasirandom groups and a Jordan type theorem. J. Eur. Math. Soc. 13, 1063–1077 (2011)

    Article  MathSciNet  Google Scholar 

  39. Oesterlé, J.: Nombres de Tamagawa et groupes unipotentes en caractéristique \(p\). Invent. Math. 78, 13–88 (1984)

    Article  MathSciNet  Google Scholar 

  40. Pyber, L., Szabó, E.: Growth in finite simple groups of Lie type. J. Amer. Math. Soc. 29, 95–146 (2016)

    Article  MathSciNet  Google Scholar 

  41. Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27, 701–717 (1980)

    Article  MathSciNet  Google Scholar 

  42. Shafarevich, I.R.: Basic Algebraic Geometry 1: Varieties in Projective Space, 3rd Russ. edn. Springer, Heidelberg (2013)

  43. Sombra, M.: A sparse effective Nullstellensatz. Adv. Appl. Math. 22, 271–295 (1999)

    Article  MathSciNet  Google Scholar 

  44. Steinberg, R.: Regular elements of semi-simple algebraic groups. Publ. Math. Inst. Hautes Études Sci. 25, 49–80 (1965)

  45. Stückrad, J., Vogel, W.: An algebraic approach to the intersection theory. Curves Semin. at Queen’s, Vol. 2, Kingston/Can. 1981-82. Queen’s Pap. Pure Appl. Math. 61, Exp. A, 32 p. (1982)

  46. Tao, T.: Expansion in Finite Simple Groups of Lie Type. Graduate Studies in Mathematics, vol. 164. American Mathematical Society, Providence, RI (2015)

  47. Tiep, P.H., Zalesskii, A.E.: Minimal characters of the finite classical groups. Commun. Algebra 24, 2093–2167 (1996)

    Article  MathSciNet  Google Scholar 

  48. Vogel, W.: Lectures on Results on Bezout’s Theorem. Notes by D. P. Patil. Lect. Math. Phys., Math., Tata Inst. Fundam. Res., vol. 74. Springer, Berlin; Tata Inst. of Fundamental Research, Bombay (1984)

  49. Wilson, R.A.: The Finite Simple Groups. Graduate Texts in Mathematics, vol. 251. Springer, London (2009)

Download references

Acknowledgements

JB was supported by ERC Consolidator grants 648329 (codename GRANT, with H. Helfgott as PI) and 681207 (codename GrDyAP, with A. Thom as PI). DD has been supported by ERC Consolidator grant 648329, the Israel Science Foundation Grants No. 686/17 and 700/21 of A. Shalev, the Emily Erskine Endowment Fund, and ERC Advanced grant 741420 (codename GROGandGIN, with L. Pyber as PI). HH was supported by ERC Consolidator grant 648329 and by his Humboldt professorship. The authors would like to thank their collaborators in Göttingen and the community of MathOverflow for suggestions and insight. Thanks are also due to Miles Reid and Dominic Bunnett for discussions on Bézout’s theorem and different meanings of the word “intersection” and to Inna Capdeboscq for helping us understand the differences between how some groups are understood in the study of linear algebraic groups and in finite group theory. Thanks are also due to two anonymous referees, in part for drawing our attention to those differences.

Author information

Authors and Affiliations

  1. Department of Mathematics, Kiel University, Kiel, 24118, Germany

    Jitendra Bajpai

  2. Alfréd Rényi Institute of Mathematics, Budapest, 1053, Hungary

    Daniele Dona

  3. Mathematisches Institut, Georg-August-Universität Göttingen, Göttingen, 37073, Germany

    Harald Andrés Helfgott

  4. IMJ-PRG, Université Paris-Cité, Boîte Courrier 7012, 8 place Aurélie Nemours, Paris Cedex 13, 75205, France

    Harald Andrés Helfgott

Authors
  1. Jitendra Bajpai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniele Dona
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Harald Andrés Helfgott
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harald Andrés Helfgott.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Dedicated to Pham Huu Tiep for his 60th birthday.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bajpai, J., Dona, D. & Helfgott, H.A. New Dimensional Estimates for Subvarieties of Linear Algebraic Groups. Vietnam J. Math. 52, 479–518 (2024). https://doi.org/10.1007/s10013-024-00687-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-024-00687-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation