AI 中文总结
研究高维Schottky空间的拓扑结构，主要定理表明在边界维数（两倍秩）时，空间的基本群是二阶循环群的乘积，但整体单连通，并证明旋转对称核是强形变收缩核。
AI 中文摘要
标记的Schottky空间记录，在共轭意义下，固定秩的自由群作为Schottky群在固定维数双曲空间上的所有作用。在三维中，它是覆盖黎曼曲面模空间的经典Schottky空间，已在复解析角度得到研究。在高维中，每个生成元获得一个旋转参数，即其轴法向方向的特殊正交变换，这在经典中没有类比。我们的主要定理处理边界维数，即两倍秩：在那里，空间的一个稠密开部分的基本群是二阶循环群的乘积，每个生成元对应一个，但整个空间是单连通的，因为每个这样的环通过最退化构型收缩。作为推论，在此边界维数中，任何两个相同秩的Schottky群是拟共形同伦的，部分回答了Kapovich的问题。我们还证明了旋转对称核在每个维数中都是强形变收缩核，该稠密开部分同伦等价于特殊正交群的乘积，并且低一维的类似轨迹有两个连通分支。
英文摘要
The marked Schottky space records, up to conjugacy, all actions of a free group of fixed rank as a Schottky group on hyperbolic space of fixed dimension. In dimension three it is the classical Schottky space covering the moduli space of Riemann surfaces, studied complex-analytically. In higher dimensions each generator gains a rotational parameter, a special orthogonal transformation of the directions normal to its axis, with no classical analogue. Our main theorem treats the borderline dimension, twice the rank: there a dense open part of the space has fundamental group a product of cyclic groups of order two, one per generator, yet the whole space is simply connected, since each such loop contracts through the most degenerate configurations. As a consequence, any two Schottky groups of the same rank in this borderline dimension are quasiconformally isotopic, partially answering a question of Kapovich. We also show that a rotationally symmetric core is a strong deformation retract in every dimension, that this dense open part is homotopy equivalent to a product of special orthogonal groups, and that the analogous locus one dimension below has two connected components.
CommentsThis paper has been withdrawn by the author. Lemma 3.1, on which the main results depend, is not established in general, and the author is unable to complete the argument at present
