标题： 代数基本群的球商的上同调非零性 显示英文标题

标题： Cohomological nonvanishing for algebraic fundamental groups of ball quotients

摘要： 假设$\Gamma < \mathrm{PU}(n,1)$是一种最简单类型的共紧致算术格，其有限群完成为$\widehat{\Gamma}$。 本文证明存在一个开子群$\widehat{\Gamma}_0 \le \widehat{\Gamma}$，使得对于每个开子群${\widehat{\Delta} \le \widehat{\Gamma}_0}$、$j \le 2n$和足够大的素数$p$，$H^j(\widehat{\Delta}, \mathbb{F}_p)$都是非平凡的。 如果$n \ge 2$，对于所有$j \ge 2$都非零。 因此，$\widehat{\Gamma}$的虚上同调维数至少为$2n$，改进了之前下限$1$的结果。 证明表明，相关的球商存在一个有限基本类，且其典范类在挠子群模下是有限的。 对于同余$\Gamma$和$j < \frac{n+1}{2}$，限制${H^j(\widehat{\Gamma}, \mathbb{F}_p) \to H^j(\Gamma, \mathbb{F}_p)}$被证明在某种精确的意义上几乎是满射的；这与格在$\mathrm{PU}(n,1)$中是否为 Serre 意义下的好格有关，这仅在$n=1$时已知成立。 显示英文摘要

摘要： Suppose $\Gamma < \mathrm{PU}(n,1)$ is a cocompact arithmetic lattice of simplest type with profinite completion $\widehat{\Gamma}$. This paper proves there is an open subgroup $\widehat{\Gamma}_0 \le \widehat{\Gamma}$ such that $H^j(\widehat{\Delta}, \mathbb{F}_p)$ is nontrivial for every open subgroup ${\widehat{\Delta} \le \widehat{\Gamma}_0}$, $j \le 2n$, and sufficiently large prime $p$. If $n \ge 2$, nonvanishing is new for all $j \ge 2$. Consequently, the virtual cohomological dimension of $\widehat{\Gamma}$ is at least $2n$, improving the previous lower bound of $1$. The proof shows there is a profinite fundamental class for the associated ball quotient and that its canonical class is profinite modulo torsion. For congruence $\Gamma$ and $j < \frac{n+1}{2}$, restriction ${H^j(\widehat{\Gamma}, \mathbb{F}_p) \to H^j(\Gamma, \mathbb{F}_p)}$ is shown to be almost surjective in a precise sense; this is related to whether lattices in $\mathrm{PU}(n,1)$ are good in the sense of Serre, which is only known to hold for $n=1$.