Zariski density and computing in arithmetic groups

Detinko, A.; Flannery, D.; Hulpke, A.

doi:10.1090/mcom/3236

Zariski density and computing in arithmetic groups
HTML articles powered by AMS MathViewer

by A. Detinko, D. L. Flannery and A. Hulpke;

Math. Comp. 87 (2018), 967-986

DOI: https://doi.org/10.1090/mcom/3236

Published electronically: August 7, 2017

PDF | Request permission

Abstract:

For $n > 2$, let $\Gamma _n$ denote either $\mathrm {SL}(n, \mathbb {Z})$ or $\mathrm {Sp}(n, \mathbb {Z})$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq \Gamma _n$. This forms the main component of our methods for computing with such arithmetic groups $H$. More generally, we provide algorithms for computing with Zariski dense groups in $\Gamma _n$. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.

References

M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI 10.1007/BF01388470
H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for $\textrm {SL}_{n}\,(n\geq 3)$ and $\textrm {Sp}_{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137. MR 244257
Bert Beisiegel, Die Automorphismengruppen homozyklischer $p$-Gruppen, Arch. Math. (Basel) 29 (1977), no. 4, 363–366 (German). MR 473016, DOI 10.1007/BF01220419
F. Beukers and G. Heckman, Monodromy for the hypergeometric function $_nF_{n-1}$, Invent. Math. 95 (1989), no. 2, 325–354. MR 974906, DOI 10.1007/BF01393900
Christopher Brav and Hugh Thomas, Thin monodromy in Sp(4), Compos. Math. 150 (2014), no. 3, 333–343. MR 3187621, DOI 10.1112/S0010437X13007550
John N. Bray, Derek F. Holt, and Colva M. Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups, London Mathematical Society Lecture Note Series, vol. 407, Cambridge University Press, Cambridge, 2013. With a foreword by Martin Liebeck. MR 3098485, DOI 10.1017/CBO9781139192576
Emmanuel Breuillard, Approximate subgroups and super-strong approximation, Groups St Andrews 2013, London Math. Soc. Lecture Note Ser., vol. 422, Cambridge Univ. Press, Cambridge, 2015, pp. 1–50. MR 3445486
Frank Celler, Charles R. Leedham-Green, Scott H. Murray, Alice C. Niemeyer, and E. A. O’Brien, Generating random elements of a finite group, Comm. Algebra 23 (1995), no. 13, 4931–4948. MR 1356111, DOI 10.1080/00927879508825509
Yao-Han Chen, Yifan Yang, and Noriko Yui, Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds, J. Reine Angew. Math. 616 (2008), 167–203. With an appendix by Cord Erdenberger. MR 2369490, DOI 10.1515/CRELLE.2008.021
Harm Derksen, Emmanuel Jeandel, and Pascal Koiran, Quantum automata and algebraic groups, J. Symbolic Comput. 39 (2005), no. 3-4, 357–371. MR 2168287, DOI 10.1016/j.jsc.2004.11.008
A. S. Detinko, D. L. Flannery, and A. Hulpke, Algorithms for arithmetic groups with the congruence subgroup property, J. Algebra 421 (2015), 234–259. MR 3272380, DOI 10.1016/j.jalgebra.2014.08.027
A. S. Detinko, D. L. Flannery, and A. Hulpke, Strong approximation and experimenting with linear groups, preprint (2016).
A. S. Detinko, D. L. Flannery, and E. A. O’Brien, Recognizing finite matrix groups over infinite fields, J. Symbolic Comput. 50 (2013), 100–109. MR 2996870, DOI 10.1016/j.jsc.2012.04.002
A. S. Detinko, D. L. Flannery, and E. A. O’Brien, Algorithms for the Tits alternative and related problems, J. Algebra 344 (2011), 397–406. MR 2831949, DOI 10.1016/j.jalgebra.2011.06.036
The GAP Group, GAP – Groups, Algorithms, and Programming, http://www.gap-system.org
Fritz Grunewald and Daniel Segal, Some general algorithms. I. Arithmetic groups, Ann. of Math. (2) 112 (1980), no. 3, 531–583. MR 595206, DOI 10.2307/1971091
Alexander J. Hahn and O. Timothy O’Meara, The classical groups and $K$-theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. MR 1007302, DOI 10.1007/978-3-662-13152-7
Jörg Hofmann and Duco van Straten, Some monodromy groups of finite index in $Sp_4(\Bbb {Z})$, J. Aust. Math. Soc. 99 (2015), no. 1, 48–62. MR 3358302, DOI 10.1017/S1446788715000014
Alexander Hulpke, Computing generators of groups preserving a bilinear form over residue class rings, J. Symbolic Comput. 50 (2013), 298–307. MR 2996881, DOI 10.1016/j.jsc.2012.08.002
A. Hulpke, The GAP package matgrp, http://www.math.colostate.edu/~hulpke/matgrp/.
James E. Humphreys, Arithmetic groups, Lecture Notes in Mathematics, vol. 789, Springer, Berlin, 1980. MR 584623
Stephen P. Humphries, Free subgroups of $\textrm {SL}(n,\textbf {Z}),\;n>2,$ generated by transvections, J. Algebra 116 (1988), no. 1, 155–162. MR 944152, DOI 10.1016/0021-8693(88)90198-6
M. Kassabov, private communication.
Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341, DOI 10.1017/CBO9780511629235
Vicente Landazuri and Gary M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418–443. MR 360852, DOI 10.1016/0021-8693(74)90150-1
D. D. Long and A. W. Reid, Small subgroups of $\textrm {SL}(3,\Bbb Z)$, Exp. Math. 20 (2011), no. 4, 412–425. MR 2859899, DOI 10.1080/10586458.2011.565258
Alexander Lubotzky, One for almost all: generation of $\textrm {SL}(n,p)$ by subsets of $\textrm {SL}(n,\textbf {Z})$, Algebra, $K$-theory, groups, and education (New York, 1997) Contemp. Math., vol. 243, Amer. Math. Soc., Providence, RI, 1999, pp. 125–128. MR 1732043, DOI 10.1090/conm/243/03689
Alexander Lubotzky, Dimension function for discrete groups, Proceedings of groups—St. Andrews 1985, London Math. Soc. Lecture Note Ser., vol. 121, Cambridge Univ. Press, Cambridge, 1986, pp. 254–262. MR 896520
Alexander Lubotzky and Chen Meiri, Sieve methods in group theory I: Powers in linear groups, J. Amer. Math. Soc. 25 (2012), no. 4, 1119–1148. MR 2947947, DOI 10.1090/S0894-0347-2012-00736-X
Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. MR 1978431, DOI 10.1007/978-3-0348-8965-0
C. R. Matthews, L. N. Vaserstein, and B. Weisfeiler, Congruence properties of Zariski-dense subgroups. I, Proc. London Math. Soc. (3) 48 (1984), no. 3, 514–532. MR 735226, DOI 10.1112/plms/s3-48.3.514
Chen Meiri, Generating pairs for finite index subgroups of $\textrm {SL}(n,\Bbb Z)$, J. Algebra 470 (2017), 420–424. MR 3565440, DOI 10.1016/j.jalgebra.2016.09.026
M. Neunhöffer, Á. Seress, et al., The GAP package recog, a collection of group recognition methods, http://gap-packages.github.io/recog/.
I. Rivin, Large Galois groups with applications to Zariski density, http://arxiv.org/abs/1312.3009v4.
Igor Rivin, Zariski density and genericity, Int. Math. Res. Not. IMRN 19 (2010), 3649–3657. MR 2725508, DOI 10.1093/imrn/rnq043
Peter Sarnak, Notes on thin matrix groups, Thin groups and superstrong approximation, Math. Sci. Res. Inst. Publ., vol. 61, Cambridge Univ. Press, Cambridge, 2014, pp. 343–362. MR 3220897
Sandip Singh, Arithmeticity of four hypergeometric monodromy groups associated to Calabi-Yau threefolds, Int. Math. Res. Not. IMRN 18 (2015), 8874–8889. MR 3417696, DOI 10.1093/imrn/rnu217
Sandip Singh and T. N. Venkataramana, Arithmeticity of certain symplectic hypergeometric groups, Duke Math. J. 163 (2014), no. 3, 591–617. MR 3165424, DOI 10.1215/00127094-2410655
G. A. Soifer and T. N. Venkataramana, Finitely generated profinitely dense free groups in higher rank semi-simple groups, Transform. Groups 5 (2000), no. 1, 93–100. MR 1745714, DOI 10.1007/BF01237181
T. N. Venkataramana, Zariski dense subgroups of arithmetic groups, J. Algebra 108 (1987), no. 2, 325–339. MR 892908, DOI 10.1016/0021-8693(87)90106-2
Thomas Weigel, On a certain class of Frattini extensions of finite Chevalley groups, Groups of Lie type and their geometries (Como, 1993) London Math. Soc. Lecture Note Ser., vol. 207, Cambridge Univ. Press, Cambridge, 1995, pp. 281–288. MR 1320526, DOI 10.1017/CBO9780511565823.019
Thomas Weigel, On the profinite completion of arithmetic groups of split type, Lois d’algèbres et variétés algébriques (Colmar, 1991) Travaux en Cours, vol. 50, Hermann, Paris, 1996, pp. 79–101. MR 1600994
A. E. Zalesskiĭ and V. N. Serežkin, Linear groups generated by transvections, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 1, 26–49, 221 (Russian). MR 412295