Cambridge University Press
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 FOR GROUP ACTIONS WITH STRONGLY CONTRACTING ELEMENTS
Cogrowth, exponential growth, divergence type, contracting element, mapping class groups, right-angled Artin groups, snowflake groups, CAT(0) groups
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Cashen, Christopher (University of Vienna)
hdl:11353/10.1086520
doi:10.1017/etds.2018.123
https://phaidra.univie.ac.at/o:1086520
Ergodic Theory and Dynamical Systems 40(7), 1738-1754 (2020-07-01)
2020-07-01