### 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.

Original language | English |
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Pages (from-to) | 4141-4149 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 145 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

### Fingerprint

### Keywords

- Bottom of spectrum
- Cogrowth
- Discrete Laplacian
- Growth tight
- Isoperimetric constant
- Planar graph
- Poincaré exponent

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*145*(10), 4141-4149. https://doi.org/10.1090/proc/13568

**Growth and cogrowth of normal subgroups of a free group.** / Jaerisch, Johannes; Matsuzaki, Katsuhiko.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 145, no. 10, pp. 4141-4149. https://doi.org/10.1090/proc/13568

}

TY - JOUR

T1 - Growth and cogrowth of normal subgroups of a free group

AU - Jaerisch, Johannes

AU - Matsuzaki, Katsuhiko

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

KW - Bottom of spectrum

KW - Cogrowth

KW - Discrete Laplacian

KW - Growth tight

KW - Isoperimetric constant

KW - Planar graph

KW - Poincaré exponent

UR - http://www.scopus.com/inward/record.url?scp=85029595740&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85029595740&partnerID=8YFLogxK

U2 - 10.1090/proc/13568

DO - 10.1090/proc/13568

M3 - Article

VL - 145

SP - 4141

EP - 4149

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 10

ER -