柱面双曲群中的增长间隙与指数一般性
Growth gaps and exponential genericity in acylindrically hyperbolic groups
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- Peking University（北京大学）
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中文总结 AI 辅助
研究柱面双曲群，证明对其每个有限生成集，非WPD元素集指数增长率更小，即WPD元素指数一般，还证明了该群的增长紧性和余增长紧性。
中文摘要 AI 辅助
我们证明，对于柱面双曲群的每个有限生成集，非WPD元素集具有严格更小的指数增长率。等价地，WPD元素是指数一般的。作为应用，我们证明了柱面双曲群的增长紧性和余增长紧性。
英文摘要
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 $n$.
