Totally Geodesic Spectra of Quaternionic Hyperbolic Orbifolds
Totally Geodesic Spectra of Quaternionic Hyperbolic Orbifolds
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四元双曲轨道的全测地谱
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发表时间：
2015
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通讯作者：
Jeffrey S. Meyer
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作者：
Jeffrey S. Meyer
本文分析和分类了有限体积四元数双曲轨道折叠的全测地子空间及其推广，由李群中的不可约格$（\mathbf{Sp}_{2n}（\mathbb{R}））^q \times \prod_{i=1}^r \mathbf{Sp}（p_i，n-p_i）\times（\mathbf{Sp}_{2n}（\mathbb{C}））^s$产生的局部对称轨道折叠.我们给出了这样一个orbifold的全测地子空间确定其可拓类的条件。本文给出了有限体积四元数双曲轨道的可测类的算术参数化，证明了四元数双曲轨道的复双曲全测地子空间决定其可测类，而真实的双曲全测地子空间则不决定其可测类.最后，我们的工具允许我们证明每个余紧格$\Gamma<\mathbf{Sp}（m，1）$，$m\ge 2$，包含拟凸曲面子群。
In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times (\mathbf{Sp}_{2n}(\mathbb{C}))^s$. We give criteria for when the totally geodesic subspaces of such an orbifold determine its commensurability class. We give a parametrization of the commensurability classes of finite volume quaternionic hyperbolic orbifolds in terms of arithmetic data, which we use to show that the complex hyperbolic totally geodesic subspaces of a quaternionic hyperbolic orbifold determine its commensurability class, but the real hyperbolic totally geodesic subspaces do not. Lastly, our tools allow us to show that every cocompact lattice $\Gamma<\mathbf{Sp}(m,1)$, $m\ge 2$, contains quasiconvex surface subgroups.