Large and Small Covers of a Hyperbolic Manifold
Large and Small Covers of a Hyperbolic Manifold
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双曲流形的大覆盖和小覆盖
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Edward C. Taylor
中科院分区：
文献类型：
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作者：
Petra Bonfert;Katsuhiko Matsuzaki;Edward C. Taylor
非初等双曲等距离散群的收敛指数表示圆锥极限集的豪斯多夫维数。在传递到非平凡正则覆盖时，所得的极限集是逐点相等的，尽管覆盖均匀化的收敛指数可能严格小于基的收敛指数。本文证明，对于封闭双曲曲面，先前建立的“小”正则覆盖收敛指数的二分之一下界是尖锐的，但不能得到。我们也考虑“大的”（非规则的）覆盖。这里的“大”和“小”描述了收敛指数的大小。我们证明了将一个流形同胚统一到一个圆上的表面纤维的Kleinian群包含一个收敛指数任意接近于2的Schottky子群。
The exponent of convergence of a non-elementary discrete group of hyperbolic isometries measures the Hausdorff dimension of the conical limit set. In passing to a non-trivial regular cover the resulting limit sets are point-wise equal though the exponent of convergence of the cover uniformization may be strictly less than the exponent of convergence of the base. We show in this paper that, for closed hyperbolic surfaces, the previously established lower bound of one half on the exponent of convergence of “small” regular covers is sharp but is not attained. We also consider “large” (non-regular) covers. Here large and small are descriptive of the size of the exponent of convergence. We show that a Kleinian group that uniformizes a manifold homeomorphic to a surface fibering over a circle contains a Schottky subgroup whose exponent of convergence is arbitrarily close to two.