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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Systolic growth of linear groups
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by Khalid Bou-Rabee and Yves Cornulier PDF
Proc. Amer. Math. Soc. 144 (2016), 529-533 Request permission


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.
  • Khalid Bou-Rabee and Tasho Kaletha, Quantifying residual finiteness of arithmetic groups, Compos. Math. 148 (2012), no. 3, 907–920. MR 2925403, DOI 10.1112/S0010437X11007469
  • Khalid Bou-Rabee and David Ben McReynolds, Extremal behavior of divisibility functions. Geometriae Dedicata, to appear. arXiv:1211.4727.
  • Khalid Bou-Rabee and Brandon Seward, Arbitrarily large residual finiteness growth. To appear in J. Reine Angew. Math.
  • Khalid Bou-Rabee and Daniel Studenmund, Full residual finiteness growths of nilpotent groups. arXiv:1406.3763 (2014), to appear in Israel J. Math.
  • Y. Cornulier. Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups. ArXiv:1403.5295 (2014).
  • 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
  • MR Author ID: 888620
  • Email:
  • Yves Cornulier
  • Affiliation: CNRS – Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France
  • MR Author ID: 766953
  • Email:
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 529-533
  • MSC (2010): Primary 20E26; Secondary 11C08, 13B25, 20F65
  • DOI:
  • MathSciNet review: 3430831