可数群的相对双曲性和相对拟凸性
Relative hyperbolicity and relative quasiconvexity for countable groups
摘要：
为研究相对双曲群的相对拟凸子群奠定了基础。这些基础要求我们首先建立可数相对双曲群（不一定是双曲生成的）的相干理论。我们证明了Gromov，Osin和Bowditch的相对双曲性可数群的定义的等价性。然后利用相对双曲群上的各种自然几何给出了相对拟凸子群的几个等价定义。我们证明了每个相对拟凸子群本身是相对双曲的，并且两个相对拟凸子群的交集也是相对拟凸的。证明了非畸变子群是相对拟凸的，并计算了非畸变子群的畸变。
英文摘要：
We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin, and Bowditch's definitions of relative hyperbolicity for countable groups. We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperbolic, and that the intersection of two relatively quasiconvex subgroups is again relatively quasiconvex. In the finitely generated case, we prove that every undistorted subgroup is relatively quasiconvex, and we compute the distortion of a finitely generated relatively quasiconvex subgroup.
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