论文标题
相对双曲线的尖端空间和准静态
Cusped spaces and quasi-isometries of relatively hyperbolic groups
论文作者
论文摘要
如果$γ$在适当的$δ$ - 双曲线空间中承认cusp-均匀的动作，则与子组$ \ mathbb {p} $的一家组$γ$相对夸张。我们表明，给定组对的任何两个这样的空间都是准等级的，只要空间具有“恒定的holosperical畸变”，这是Groves满足的条件 - Manning的Cused Cayley图和所有负弯曲的对称空间。因此，Bowditch边界承认了一个规范的准对称结构，该结构与对称空间的“天然发生”的准对称结构相吻合时，考虑了等级一级对称空间中的晶格。 我们表明，当$γ$的$γ$是一个负弯曲的对称空间中的晶格$ x $，并且仅当$γ$的cused空间与对称空间是准时的。我们还证明了$δ$的理想三角特征 - 由于Meyer和Bourdon-Kleiner，具有均匀完美边界的双曲线空间。附录涉及文献中发现的几个锥形极限点定义的等效性。
A group $Γ$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $Γ$ admits a cusp-uniform action on a proper $δ$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric, provided the spaces have "constant horospherical distortion," a condition satisfied by Groves--Manning's cusped Cayley graph and by all negatively curved symmetric spaces. Consequently the Bowditch boundary admits a canonical quasisymmetric structure, which coincides with the "naturally occurring" quasisymmetric structure of the symmetric space when considering lattices in rank one symmetric spaces. We show that a group $Γ$ is a lattice in a negatively curved symmetric space $X$ if and only if a cusped space for $Γ$ is quasi-isometric to the symmetric space. We also prove an ideal triangle characterization of the $δ$--hyperbolic spaces with uniformly perfect boundary due to Meyer and Bourdon--Kleiner. An appendix concerns the equivalence of several definitions of conical limit point found in the literature.
