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 δN and δG 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 δN /δG > 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 metrics and a recent result of Jaerisch, Matsuzaki, and Yabuki on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

Cogrowth, exponential growth, divergence type, contracting element, mapping class groups, right-angled Artin groups, snowflake groups, CAT(0) groups

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Ergodic Theory and Dynamical Systems

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Cambridge University Press