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Growth gaps and exponential genericity in acylindrically hyperbolic groups
Abstract
Let G be a finitely generated group acting by isometries on a geodesic metric space with two independent strongly contracting WPD elements. We prove that, for every finite generating set of G, strongly contracting WPD elements for this action are exponentially generic in word-metric balls. The proof establishes a growth gap for elements with small displacement relative to word length in an auxiliary projection complex. For finitely generated acylindrically hyperbolic groups, we establish growth tightness and cogrowth tightness for every finite generating set. We also show that the number of conjugacy classes meeting a ball of radius n is comparable to the cardinality of that ball divided by n.
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BibTeX
@misc{ding2026growth,
title = {{Growth gaps and exponential genericity in acylindrically hyperbolic groups}},
author = {Lihuang Ding and Wenyuan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.25578},
eprint = {2607.25578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.25578},
}
RIS
TY - UNPB
TI - Growth gaps and exponential genericity in acylindrically hyperbolic groups
AU - Lihuang Ding
AU - Wenyuan Yang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2607.25578
UR - https://arxiv.org/abs/2607.25578
AB - Let G be a finitely generated group acting by isometries on a geodesic metric space with two independent strongly contracting WPD elements. We prove that, for every finite generating set of G, strongly contracting WPD elements for this action are exponentially generic in word-metric balls. The proof establishes a growth gap for elements with small displacement relative to word length in an auxiliary projection complex. For finitely generated acylindrically hyperbolic groups, we establish growth tightness and cogrowth tightness for every finite generating set. We also show that the number of conjugacy classes meeting a ball of radius n is comparable to the cardinality of that ball divided by n.
ER -