论文标题
双曲线组的增长率
The rates of growth in a hyperbolic group
论文作者
论文摘要
我们研究双曲线群体相对于其所有有限产生集的可计数增长率集。我们证明该集合是有序的，每个实际数字都可以是最多有限的许多生成的增长速率，这是该组的自动形态。我们证明，增长率集的序列至少为$ω^ω$，如果该组是极限组（例如，自由和表面组），则为$ω^ω$。 我们进一步研究双曲线组的所有有限产生的亚组的增长率相对于它们的所有有限产生集。事实证明，该集合也是井井有条的，每个实际数字都可以是最多有限的许多同构类别的有限同构生成的生成一组的子组的子组的增长率。最后，我们加强了结果，以包括双曲线组所有次元的所有有限产生集的增长率。
We study the countable set of rates of growth of a hyperbolic group with respect to all its finite generating sets. We prove that the set is well-ordered, and that every real number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. We prove that the ordinal of the set of rates of growth is at least $ω^ω$, and in case the group is a limit group (e.g., free and surface groups), it is $ω^ω$. We further study the rates of growth of all the finitely generated subgroups of a hyperbolic group with respect to all their finite generating sets. This set is proved to be well-ordered as well, and every real number can be the rate of growth of at most finitely many isomorphism classes of finite generating sets of subgroups of a given hyperbolic group. Finally, we strengthen our results to include rates of growth of all the finite generating sets of all the subsemigroups of a hyperbolic group.
