Extremal behavior of divisibility functions

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• 作者：Khalid Bou-Rabee (1)
  D. B. McReynolds (2)
  
  1. University of Michigan ; Ann Arbor ; MI ; 48109 ; USA
  2. Purdue University ; West Lafayette ; IN ; 47907 ; USA
  
• 关键词：Residual finiteness ; Linear groups ; Divisibility functions ; Nilpotent groups ; 20E26 ; 20G05
• 刊名：Geometriae Dedicata
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：175
• 期：1
• 页码：407-415
• 全文大小：149 KB
• 参考文献：1. Bou-Rabee, K (2010) Quantifying residual finiteness. J. Algebra 323: pp. 729-737 CrossRef
  2. Bou-Rabee, K (2011) Approximating a group by its solvable quotients. N. Y. J. Math. 17: pp. 699-712
  3. Bou-Rabee, K, Kaletha, T (2012) Quantifying residual finiteness of arithmetic groups. Compos. Math. 148: pp. 907-920 CrossRef
  4. Bou-Rabee, K, McReynolds, DB (2010) Bertrand鈥檚 postulate and subgroup growth. J. Algebra 324: pp. 793-819 CrossRef
  5. Bou-Rabee, K, McReynolds, DB (2011) Asymptotic growth and least common multiples in groups. Bull. Lond. Math. Soc. 43: pp. 1059-1068 CrossRef
  6. Kassabov, M, Matucci, F (2011) Bounding the residual finiteness of free groups. Proc. Am. Math. Soc. 139: pp. 2281-2286 CrossRef
  7. Lubtozky, A, Segal, D (2003) Subgroup Growth. Birkh盲user, Basel CrossRef
  8. Mal鈥檆ev, AI (1940) On the faithful representation of infinite groups by matrices. Mat. SS. (N.S.) 50: pp. 405-422
  9. Roman, S (1995) Field Theory. Springer, Berlin CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9168

文摘

In this short article, we study the extremal behavior \({{\mathrm{F}}}_\Gamma (n)\) of divisibility functions \({{\mathrm{D}}}_\Gamma \) introduced by the first author for finitely generated groups \(\Gamma \) . These functions aim at quantifying residual finiteness and bounds give a measurement of the complexity in verifying a word is non-trivial. We show that finitely generated subgroups of \({{\mathrm{GL}}}(m,K)\) for an infinite field \(K\) have at most polynomial growth for the function \({{\mathrm{F}}}_\Gamma (n)\) . Consequently, we obtain a dichotomy for the growth rate of \(\log {{\mathrm{F}}}_\Gamma (n)\) for finitely generated subgroups of \({{\mathrm{GL}}}(n,\mathbf {C})\) . We also show that if \({{\mathrm{F}}}_\Gamma (n) \preceq \log \log n\) , then \(\Gamma \) is finite. In contrast, when \(\Gamma \) contains an element of infinite order, \(\log n \preceq {{\mathrm{F}}}_\Gamma (n)\) . We end with a brief discussion of some geometric motivation for this work.