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Cogrowth for group actions with strongly contracting elements
Published online by Cambridge University Press: 04 December 2018
Abstract
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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.
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