论文标题
算术局部对称空间的同型类型和同源性与同源量
Homotopy type and homology versus volume for arithmetic locally symmetric spaces
论文作者
论文摘要
我们研究了半岛谎言组中与算术晶格相关的局部对称空间。我们证明了以下有关其拓扑结构的结果：三角剖分所需的四面体数量最少在体积中是线性的，并且贝蒂数在体积中是亚线性的，除了可能的中间程度。这些结果的证据使用了这些空间的几何形状，即对它们的薄部分的研究。在这方面，我们证明，这些空间在本杰明·塞拉姆（Benjamini-Schramm）的感觉中融合到它们的通用覆盖物中，并为较大程度的痕量轨道的薄零件的体积明确绑定。我们证明的主要技术成分是对轨道积分的新估计值，小位移元素的计数结果以及算术局部对称空间的Margulis引理的精制版本。
We study locally symmetric spaces associated with arithmetic lattices in semisimple Lie groups. We prove the following results about their topology: the minimal number of tetrahedra needed for a triangulation is at most linear in the volume and the Betti numbers are sub-linear in the volume except possibly in middle degree. The proof of these results uses the geometry of these spaces, namely the study of their thin parts. In this regard we prove that these spaces converge in the Benjamini--Schramm sense to their universal covers and give an explicit bound for the volume of the thin part for trace fields of large degree. The main technical ingredients for our proofs are new estimates on orbital integrals, a counting result for elements of small displacement, and a refined version of the Margulis lemma for arithmetic locally symmetric spaces.
