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Growth and cogrowth of normal subgroups of a free group


Authors: Johannes Jaerisch and Katsuhiko Matsuzaki
Journal: Proc. Amer. Math. Soc. 145 (2017), 4141-4149
MSC (2010): Primary 20F69, 05C50; Secondary 20E08, 30F40
DOI: https://doi.org/10.1090/proc/13568
Published electronically: May 4, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a sufficient condition for a sequence of normal subgroups of a free group to have the property that both their growths tend to the upper bound and their cogrowths tend to the lower bound. The condition is represented by planarity of the quotient graphs of the tree.


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  • [AL02] G. N. Arzhantseva and I. G. Lysenok, Growth tightness for word hyperbolic groups, Math. Z. 241 (2002), no. 3, 597-611. MR 1938706, https://doi.org/10.1007/s00209-002-0434-6
  • [Coh82] Joel M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), no. 3, 301-309. MR 678175, https://doi.org/10.1016/0022-1236(82)90090-8
  • [FS04] Kurt Falk and Bernd O. Stratmann, Remarks on Hausdorff dimensions for transient limit sets of Kleinian groups, Tohoku Math. J. (2) 56 (2004), no. 4, 571-582. MR 2097162
  • [Fuj96] Koji Fujiwara, Growth and the spectrum of the Laplacian of an infinite graph, Tohoku Math. J. (2) 48 (1996), no. 2, 293-302. MR 1387821, https://doi.org/10.2748/tmj/1178225382
  • [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, https://doi.org/10.1007/BF02471762
  • [Gri80] R. I. Grigorchuk, Symmetrical random walks on discrete groups, Multicomponent random systems, Adv. Probab. Related Topics, vol. 6, Dekker, New York, 1980, pp. 285-325. MR 599539
  • [Jae15] Johannes Jaerisch, A lower bound for the exponent of convergence of normal subgroups of Kleinian groups, J. Geom. Anal. 25 (2015), no. 1, 298-305. MR 3299281, https://doi.org/10.1007/s12220-013-9427-4
  • [Jae16] Johannes Jaerisch, Recurrence and pressure for group extensions, Ergodic Theory Dynam. Systems 36 (2016), no. 1, 108-126. MR 3436756, https://doi.org/10.1017/etds.2014.54
  • [Kes59a] Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146-156. MR 0112053
  • [Kes59b] Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336-354. MR 0109367
  • [Mas88] Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
  • [Moh88] Bojan Mohar, Isoperimetric inequalities, growth, and the spectrum of graphs, Linear Algebra Appl. 103 (1988), 119-131. MR 943998, https://doi.org/10.1016/0024-3795(88)90224-8
  • [MW89] Bojan Mohar and Wolfgang Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989), no. 3, 209-234. MR 986363, https://doi.org/10.1112/blms/21.3.209
  • [MYJ15] K. Matsuzaki, Y. Yabuki, and J. Jaerisch, Normalizer, divergence type and Patterson measure for discrete groups of the Gromov hyperbolic space, arXiv:1511.02664 (2015).
  • [Rob05] Thomas Roblin, Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math. 147 (2005), 333-357 (French, with French summary). MR 2166367, https://doi.org/10.1007/BF02785371
  • [Shu99] A. G. Shukhov, On the dependence of the growth exponent on the length of the defining relation, Mat. Zametki 65 (1999), no. 4, 612-618 (Russian, with Russian summary); English transl., Math. Notes 65 (1999), no. 3-4, 510-515. MR 1715061, https://doi.org/10.1007/BF02675367
  • [Sul87] Dennis Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), no. 3, 327-351. MR 882827
  • [Wag37] K. Wagner, Über eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), no. 1, 570-590 (German). MR 1513158, https://doi.org/10.1007/BF01594196

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Additional Information

Johannes Jaerisch
Affiliation: Shimane University, Nishi-Kawatsu-cho 1060, Matsue, Shimane 690-8504, Japan

Katsuhiko Matsuzaki
Affiliation: Department of Mathematics, School of Education, Waseda University, Nishi-Waseda 1-6-1, Shiujuku, Tokyo 169-8050, Japan

DOI: https://doi.org/10.1090/proc/13568
Keywords: Growth tight, cogrowth, Poincar\'e exponent, discrete Laplacian, bottom of spectrum, isoperimetric constant, planar graph
Received by editor(s): December 15, 2015
Received by editor(s) in revised form: September 30, 2016, and October 20, 2016
Published electronically: May 4, 2017
Communicated by: Nimish A. Shah
Article copyright: © Copyright 2017 American Mathematical Society

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