Tameness of Margulis space-times with parabolics
Tameness of Margulis space-times with parabolics
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DOI：
10.1515/forum-2019-0331
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发表时间：
2017-10
影响因子：
0.8
通讯作者：
Suhyoung Choi;Todd A. Drumm;W. Goldman
中科院分区：
文献类型：
--
作者：
Suhyoung Choi;Todd A. Drumm;W. Goldman
摘要 设 𝖤 {\mathsf{E}} 为签名为 ( 2 , 1 ) {(2,1)} 的平坦洛伦兹空间。 Margulis 时空是一个非紧完备洛伦兹平面 3 流形 𝖤 / Γ {\mathsf{E}/\Gamma} ，具有秩为 𝗀 {\mathsf{g}} 、𝗀 ≥ 2 {\mathsf{g}\geq 2} 的自由完整群 Γ 。我们考虑 Γ 包含抛物线元素的情况。我们根据 Margulis 和 Charette-Drumm 不变量获得了适当的 Γ 作用的表征。我们证明 𝖤 / Γ {\mathsf{E}/\Gamma} 与属 𝗀 {\mathsf{g}} 的紧凑手柄内部同胚，概括了我们之前的结果。此外，我们通过在无穷远处添加实射影面，获得了用抛物线对 Margulis 时空进行的边界化，从而将我们压缩为相对于抛物线端邻域的流形。我们的方法是估计仿射变换群的平移部分并使用一些三流形拓扑。
Abstract Let 𝖤 {\mathsf{E}} be a flat Lorentzian space of signature ( 2 , 1 ) {(2,1)} . A Margulis space-time is a noncompact complete Lorentz flat 3-manifold 𝖤 / Γ {\mathsf{E}/\Gamma} with a free holonomy group Γ of rank 𝗀 {\mathsf{g}} , 𝗀 ≥ 2 {\mathsf{g}\geq 2} . We consider the case when Γ contains a parabolic element. We obtain a characterization of proper Γ-actions in terms of Margulis and Charette–Drumm invariants. We show that 𝖤 / Γ {\mathsf{E}/\Gamma} is homeomorphic to the interior of a compact handlebody of genus 𝗀 {\mathsf{g}} generalizing our earlier result. Also, we obtain a bordification of the Margulis space-time with parabolics by adding a real projective surface at infinity giving us a compactification as a manifold relative to parabolic end neighborhoods. Our method is to estimate the translational parts of the affine transformation group and use some 3-manifold topology.