@article{474a3dd1b20047c99587cd00e3a4cc0a,
title = "Growth and cogrowth of normal subgroups of a free group",
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.",
keywords = "Bottom of spectrum, Cogrowth, Discrete Laplacian, Growth tight, Isoperimetric constant, Planar graph, Poincar{\'e} exponent",
author = "Johannes Jaerisch and Katsuhiko Matsuzaki",
note = "Funding Information: The authors would like to thank Wenyuan Yang for introducing the growth tightness of (relatively) hyperbolic groups to them. The authors would like to thank the referee for the careful reading of the manuscript. The research of the first author was partially supported by the JSPS Postdoctoral Fellowship P14321 and the JSPS KAKENHI 15H06416. The research of the second author was partially supported by the JSPS KAKENHI 16K13767. Publisher Copyright: {\textcopyright} 2017 American Mathematical Society.",
year = "2017",
doi = "10.1090/proc/13568",
language = "English",
volume = "145",
pages = "4141--4149",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",
number = "10",
}