2 results
Patterson–Sullivan theory for groups with a strongly contracting element
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 11 / November 2024
- Published online by Cambridge University Press:
- 05 March 2024, pp. 3216-3271
- Print publication:
- November 2024
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Cogrowth for group actions with strongly contracting elements
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 7 / July 2020
- Published online by Cambridge University Press:
- 04 December 2018, pp. 1738-1754
- Print publication:
- July 2020
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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.
