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.
Introduction to a small cancellation theorem
