Intersection growth in groups

Biringer, Ian; Bou-Rabee, Khalid; Kassabov, Martin; Matucci, Francesco

doi:10.1090/tran/6865

Intersection growth in groups

Authors: Ian Biringer, Khalid Bou-Rabee, Martin Kassabov and Francesco Matucci
Journal: Trans. Amer. Math. Soc. 369 (2017), 8343-8367
MSC (2010): Primary 20F69; Secondary 20E05, 20E07, 20E26, 20E28
DOI: https://doi.org/10.1090/tran/6865
Published electronically: August 22, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The intersection growth of a group is the asymptotic behavior of the index of the intersection of all subgroups of with index at most , and measures the Hausdorff dimension of in profinite metrics. We study intersection growth in free groups and special linear groups and relate intersection growth to quantifying residual finiteness.

[BMS67] H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for 𝑆𝐿_{𝑛}(𝑛≥3) and 𝑆𝑝_{2𝑛}(𝑛≥2), Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137. MR 0244257
[Bou10] Khalid Bou-Rabee, Quantifying residual finiteness, J. Algebra 323 (2010), no. 3, 729–737. MR 2574859, https://doi.org/10.1016/j.jalgebra.2009.10.008
[Bou11] Khalid Bou-Rabee, Approximating a group by its solvable quotients, New York J. Math. 17 (2011), 699–712. MR 2851069
[BM10] K. Bou-Rabee and D. B. McReynolds, Bertrand’s postulate and subgroup growth, J. Algebra 324 (2010), no. 4, 793–819. MR 2651569, https://doi.org/10.1016/j.jalgebra.2010.03.031
[BM11] K. Bou-Rabee and D. B. McReynolds, Asymptotic growth and least common multiples in groups, Bull. Lond. Math. Soc. 43 (2011), no. 6, 1059–1068. MR 2861528, https://doi.org/10.1112/blms/bdr038
[BM12] Khalid Bou-Rabee and D. B. McReynolds, Extremal behavior of divisibility functions, Geom. Dedicata 175 (2015), 407–415. MR 3323650, https://doi.org/10.1007/s10711-014-9955-5
[BK12] Khalid Bou-Rabee and Tasho Kaletha, Quantifying residual finiteness of arithmetic groups, Compos. Math. 148 (2012), no. 3, 907–920. MR 2925403, https://doi.org/10.1112/S0010437X11007469
[Bus09] N. V. Buskin, Efficient separability in free groups, Sibirsk. Mat. Zh. 50 (2009), no. 4, 765–771 (Russian, with Russian summary); English transl., Sib. Math. J. 50 (2009), no. 4, 603–608. MR 2583614, https://doi.org/10.1007/s11202-009-0067-7
[CCNPW85] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
[C78] Bruce N. Cooperstein, Minimal degree for a permutation representation of a classical group, Israel J. Math. 30 (1978), no. 3, 213–235. MR 0506701, https://doi.org/10.1007/BF02761072
[GKP12] F. Grunewald, B.Kunyavskii, and E. Plotkin, New trends in characterization of solvable groups, to appear.
[Had11] Uzy Hadad, On the shortest identity in finite simple groups of Lie type, J. Group Theory 14 (2011), no. 1, 37–47. MR 2764921, https://doi.org/10.1515/JGT.2010.039
[Hall59] Marshall Hall Jr., The theory of groups, The Macmillan Co., New York, N.Y., 1959. MR 0103215
[His84] Gerhard Hiss, Die adjungierten Darstellungen der Chevalley-Gruppen, Arch. Math. (Basel) 42 (1984), no. 5, 408–416 (German). MR 756692, https://doi.org/10.1007/BF01190689
[Hog82] G. M. D. Hogeweij, Almost-classical Lie algebras. I, II, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 4, 441–452, 453–460. MR 683531
[Jon74] Gareth A. Jones, Varieties and simple groups, J. Austral. Math. Soc. 17 (1974), 163–173. Collection of articles dedicated to the memory of Hanna Neumann, VI. MR 0344342
[KL90] William M. Kantor and Alexander Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), no. 1, 67–87. MR 1065213, https://doi.org/10.1007/BF00181465
[KM11] Martin Kassabov and Francesco Matucci, Bounding the residual finiteness of free groups, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2281–2286. MR 2784792, https://doi.org/10.1090/S0002-9939-2011-10967-5
[KMS12] O. Kharlampovich, A. Myasnikov, and M. Sapir, Algorithmically complex residually finite groups,
submitted, http://arxiv.org/abs/1204.6506.
[KN66] L. G. Kovács and M. F. Newman, Cross varieties of groups, Proc. Roy. Soc. Ser. A 292 (1966), 530–536. MR 0194505
[KT15] Gady Kozma and Andreas Thom, Divisibility and laws in finite simple groups, Math. Ann. 364 (2016), no. 1-2, 79–95. MR 3451381, https://doi.org/10.1007/s00208-015-1201-4
[LS03] Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. MR 1978431
[LS95] Martin W. Liebeck and Aner Shalev, The probability of generating a finite simple group, Geom. Dedicata 56 (1995), no. 1, 103–113. MR 1338320, https://doi.org/10.1007/BF01263616
[KK00] Luise-Charlotte Kappe and Joseph Kirtland, Supplementation in groups, Glasg. Math. J. 42 (2000), no. 1, 37–50. MR 1739693, https://doi.org/10.1017/S0017089500010065
[MP94] Allen Moy and Gopal Prasad, Unrefined minimal 𝐾-types for 𝑝-adic groups, Invent. Math. 116 (1994), no. 1-3, 393–408. MR 1253198, https://doi.org/10.1007/BF01231566
[MQR13] Nina E. Menezes, Martyn Quick, and Colva M. Roney-Dougal, The probability of generating a finite simple group, Israel J. Math. 198 (2013), no. 1, 371–392. MR 3096644, https://doi.org/10.1007/s11856-013-0034-7
[Neu67] Hanna Neumann, Varieties of groups, Springer-Verlag New York, Inc., New York, 1967. MR 0215899
[NS07] Nikolay Nikolov and Dan Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds, Ann. of Math. (2) 165 (2007), no. 1, 171–238. MR 2276769, https://doi.org/10.4007/annals.2007.165.171
[N87] Madhav V. Nori, On subgroups of 𝐺𝐿_{𝑛}(𝐹_{𝑝}), Invent. Math. 88 (1987), no. 2, 257–275. MR 880952, https://doi.org/10.1007/BF01388909
[OP64] Sheila Oates and M. B. Powell, Identical relations in finite groups, J. Algebra 1 (1964), 11–39. MR 0161904, https://doi.org/10.1016/0021-8693(64)90004-3
[PR95] Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
[Wil09] Robert A. Wilson, The finite simple groups, Graduate Texts in Mathematics, vol. 251, Springer-Verlag London, Ltd., London, 2009. MR 2562037