大临界指数的自由半群
Free Semigroups of Large Critical Exponent
AI总结：
本文通过构造Bishop-Jones半群，证明对于带有扩张粗上循环的收敛群，存在临界指数任意接近但严格小于环境群临界指数的有限生成自由子半群，并应用于半单Lie群的横截子群得到Anosov子半群。
摘要：
对于配备扩张粗上循环的收敛群，我们构造了有限生成的自由子半群，称为Bishop-Jones半群，其临界指数任意接近但严格小于环境群的临界指数。作为应用，我们证明对于半单Lie群$G$的任意非初等横截子群$\Gamma$，存在Kassel-Potrie意义下的有限生成自由Anosov子半群，其临界指数任意接近但严格小于环境横截群的临界指数。此外，我们证明这些半群允许等距嵌入到$G$的对称空间$X$中，并具有某些额外的粗几何性质。
英文摘要：
For a convergence group equipped with an expanding coarse-cocycle, we construct finitely generated free subsemigroups, which we call Bishop--Jones semigroups, of critical exponent arbitrarily close to but strictly less than the critical exponent of the ambient group. As an application, we show that for any non-elementary transverse subgroup $Γ$ of a semisimple Lie group $G$, there exist finitely generated free Anosov subsemigroups in the sense of Kassel--Potrie of critical exponent arbitrarily close to but strictly less than that of the ambient transverse group. Furthermore, we show that these semigroups admit quasi-isometric embeddings into the symmetric space X of G with certain additional coarse-geometric properties.
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