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Growth of quasiconvex subgroups

Published online by Cambridge University Press:  28 June 2018

FRANÇOIS DAHMANI ,
DAVID FUTER  and
DANIEL T. WISE
FRANÇOIS DAHMANI
Affiliation:
Université Grenoble Alpes, Institut Fourier (UMR 5582), 100 Rue des Maths, CS 40700. F-38 058 Grenoble, Cedex 9, France. e-mail: franç[email protected]
DAVID FUTER
Affiliation:
Dept. of Mathematics, Temple University, Philadelphia, PA 19122U.S.A. e-mail: [email protected]
DANIEL T. WISE
Affiliation:
Dept. of Math. & Stats. McGill University Montreal, QC, CanadaH3A 0B9 e-mail: [email protected]
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Abstract

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron–Frobenius theory.

We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel and Osin on rotating families.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 3 , November 2019 , pp. 505 - 530
Copyright
Copyright © Cambridge Philosophical Society 2018 

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