Generalising results of Razborov and Safin, and answering a question of Button, we prove that for every hyperbolic group there exists a constant $\alpha >0$ such that for every finite subset $U$ that is not contained in a virtually cyclic subgroup $|U^n{|⩾(\alpha |U|)}^{[(n+1)/2]}$. Similar estimates are established for groups acting acylindrically on trees or hyperbolic spaces.

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产品组的增长和双曲几何

推导Razborov和Safin的结果，并回答Button问题，我们证明对于每个双曲组都存在一个常数 $\alpha >0$ 这样对于每个有限子集 $ü$ 不包含在虚拟循环子集中 $|{ü}^{ñ}{|⩾（\alpha |ü|）}^{[（ñ+\mathrm{1个}）/2]}$。对于在树或双曲空间上呈圆柱状作用的组，也建立了类似的估计。