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Systolic growth of linear groups


Authors: Khalid Bou-Rabee and Yves Cornulier
Journal: Proc. Amer. Math. Soc. 144 (2016), 529-533
MSC (2010): Primary 20E26; Secondary 11C08, 13B25, 20F65
DOI: https://doi.org/10.1090/proc12747
Published electronically: June 30, 2015
MathSciNet review: 3430831
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Abstract: We prove that the residual girth of any finitely generated linear group is at most exponential. This means that the smallest finite quotient in which the $ n$-ball injects has at most exponential size. If the group is also not virtually nilpotent, it follows that the residual girth and the systolic growth are precisely exponential.


References [Enhancements On Off] (What's this?)

  • [BK] 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
  • [BM] Khalid Bou-Rabee and David Ben McReynolds, Extremal behavior of divisibility functions. Geometriae Dedicata, to appear. arXiv:1211.4727.
  • [BSe] Khalid Bou-Rabee and Brandon Seward, Arbitrarily large residual finiteness growth. To appear in J. Reine Angew. Math.
  • [BSt] Khalid Bou-Rabee and Daniel Studenmund, Full residual finiteness growths of nilpotent groups. arXiv:1406.3763 (2014), to appear in Israel J. Math.
  • [C] Y. Cornulier. Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups. ArXiv:1403.5295 (2014).
  • [G] Mikhael Gromov, Systoles and intersystolic inequalities, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 291–362 (English, with English and French summaries). MR 1427763

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Additional Information

Khalid Bou-Rabee
Affiliation: The City College of New York, 160 Convent Ave, New York, New York 10031
Email: khalid.math@gmail.com

Yves Cornulier
Affiliation: CNRS – Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France
Email: yves.cornulier@math.u-psud.fr

DOI: https://doi.org/10.1090/proc12747
Received by editor(s): August 28, 2014
Received by editor(s) in revised form: February 3, 2015
Published electronically: June 30, 2015
Additional Notes: The first-named author was supported in part by NSF DMS-1405609
The second-named author was supported in part by ANR GSG 12-BS01-0003-01
Communicated by: Kevin Whyte
Article copyright: © Copyright 2015 American Mathematical Society