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Uniform growth rate


Authors: Kasra Rafi and Jing Tao
Journal: Proc. Amer. Math. Soc. 144 (2016), 1415-1427
MSC (2010): Primary 20F36, 20F65, 57M07
DOI: https://doi.org/10.1090/proc/12816
Published electronically: December 22, 2015
MathSciNet review: 3451220
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Abstract: In an evolutionary system in which the rules of mutation are local in nature, the number of possible outcomes after $ m$ mutations is an exponential function of $ m$ but with a rate that depends only on the set of rules and not the size of the original object. We apply this principle to find a uniform upper bound for the growth rate of certain groups including the mapping class group. We also find a uniform upper bound for the growth rate of the number of homotopy classes of triangulations of an oriented surface that can be obtained from a given triangulation using $ m$ diagonal flips.


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

  • [AP14] Jayadev S. Athreya and Amritanshu Prasad,
    Growth in right-angled groups and monoids,
    Preprint, 2014,
    Available at arXiv:1409.4142 [math.GR].
  • [Art47] E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101-126. MR 0019087 (8,367a)
  • [Bir74] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. MR 0375281 (51 #11477)
  • [FM12] Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125 (2012h:57032)
  • [GdlH97] R. Grigorchuk and P. de la Harpe, On problems related to growth, entropy, and spectrum in group theory, J. Dynam. Control Systems 3 (1997), no. 1, 51-89. MR 1436550 (98d:20039), https://doi.org/10.1007/BF02471762
  • [HM95] Susan Hermiller and John Meier, Algorithms and geometry for graph products of groups, J. Algebra 171 (1995), no. 1, 230-257. MR 1314099 (96a:20052), https://doi.org/10.1006/jabr.1995.1010
  • [Lic64] W. B. R. Lickorish, A finite set of generators for the homeotopy group of a $ 2$-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769-778. MR 0171269 (30 #1500)
  • [LS77] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064 (58 #28182)
  • [Mac03] David J. C. MacKay, Information theory, inference and learning algorithms, Cambridge University Press, New York, 2003. MR 2012999 (2004i:94001)
  • [McM14] Curtis T. McMullen, Entropy and the clique polynomial, J. Topol. 8 (2015), no. 1, 184-212. MR 3335252, https://doi.org/10.1112/jtopol/jtu022
  • [MKS66] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. MR 0207802 (34 #7617)
  • [RT13] Kasra Rafi and Jing Tao, The diameter of the thick part of moduli space and simultaneous Whitehead moves, Duke Math. J. 162 (2013), no. 10, 1833-1876. MR 3079261, https://doi.org/10.1215/00127094-2323128
  • [Sco07] Richard Scott, Growth series for vertex-regular CAT(0) cube complexes, Algebr. Geom. Topol. 7 (2007), 285-300. MR 2308945 (2008e:20065), https://doi.org/10.2140/agt.2007.7.285
  • [STT92] Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston, Short encodings of evolving structures, SIAM J. Discrete Math. 5 (1992), no. 3, 428-450. MR 1172751 (93m:68110), https://doi.org/10.1137/0405034

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Additional Information

Kasra Rafi
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada
Email: rafi@math.toronto.edu

Jing Tao
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-0315
Email: jing@math.ou.edu

DOI: https://doi.org/10.1090/proc/12816
Received by editor(s): August 28, 2014
Received by editor(s) in revised form: April 9, 2015
Published electronically: December 22, 2015
Additional Notes: The first author was partially supported by NCERC Research Grant, RGPIN 435885.
The second author was partially supported by NSF Research Grant, DMS-1311834
Communicated by: Kevin Whyte
Article copyright: © Copyright 2015 American Mathematical Society

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