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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
Published electronically: May 4, 2017
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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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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

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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