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Cogrowth for group actions with strongly contracting elements

Part of: Smooth dynamical systems: general theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  04 December 2018

GOULNARA N. ARZHANTSEVA  and
CHRISTOPHER H. CASHEN Open the ORCID record for CHRISTOPHER H. CASHEN [Opens in a new window]
GOULNARA N. ARZHANTSEVA
Affiliation:
Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Österreich email goulnara.arzhantseva@univie.ac.at, christopher.cashen@univie.ac.at
CHRISTOPHER H. CASHEN
Affiliation:
Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Österreich email goulnara.arzhantseva@univie.ac.at, christopher.cashen@univie.ac.at
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Abstract

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$ and let $\unicode[STIX]{x1D6FF}_{N}$ and $\unicode[STIX]{x1D6FF}_{G}$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric, then $\unicode[STIX]{x1D6FF}_{N}/\unicode[STIX]{x1D6FF}_{G}>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk’s original result on free groups with respect to a word metric and a recent result of Matsuzaki, Yabuki and Jaerisch on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

Keywords

cogrowthexponential growthcontracting elementmapping class groupsCAT(0) groups

MSC classification

Primary: 20F69: Asymptotic properties of groups 37C35: Orbit growth
Secondary: 20F65: Geometric group theory 20F67: Hyperbolic groups and nonpositively curved groups 20F36: Braid groups; Artin groups
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1738 - 1754
Copyright
© Cambridge University Press, 2018

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