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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 (99a:53051)

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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

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