AI 中文总结
该研究通过支架泊松过程构造高阶局部对称空间的随机开覆盖，证明高阶对称空间对应无挠格序列的一阶同调归一化挠率对数与体积比值趋于0，回答了相关问题并证实了对应猜想，还得到了定量界及仿射建筑的类似结论。
AI 中文摘要
我们利用一种称为支架泊松过程（scaffolded Poisson processes）的构造，构造高阶局部对称空间的随机开覆盖。设X=G/K为非紧型且实秩至少为2的对称空间，我们对G中无挠格序列的一阶同调中归一化挠率证明了一个一般消失定理。特别地，若G是单的，对任意不同流形序列Mₙ=Γₙ\backslash X，有log|H₁(Mₙ;ℤ)_{tors}|/vol(Mₙ)→0。这回答了阿贝尔特（Abért）、杰伦德（Gelander）和尼科洛夫（Nikolov）提出的问题，并证实了伯杰龙（Bergeron）与文卡特什（Venkatesh）猜想在高阶情形下关于平凡整系数的一阶消失结论。此外，我们得到了一阶同调挠率及Γ的最小生成元个数相对于内射半径下界的定量界。最后，我们对仿射建筑证明了类似结论。
英文摘要
We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = Γ_n \backslash X$. This answers a question of Abért, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $Γ$. Finally, we prove the analogous statements for affine buildings.
Comments19 pages, minor changes in the abstract and introduction
