Cohomological nonvanishing for algebraic fundamental groups of ball quotients
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909758081990656 |
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| author | Stover, Matthew |
| author_facet | Stover, Matthew |
| contents | Suppose $Γ< \mathrm{PU}(n,1)$ is a cocompact arithmetic lattice of simplest type with profinite completion $\widehatΓ$. This paper proves there is an open subgroup $\widehatΓ_0 \le \widehatΓ$ such that $H^j(\widehatΔ, \mathbb{F}_p)$ is nontrivial for every open subgroup ${\widehatΔ \le \widehatΓ_0}$, $j \le 2n$, and sufficiently large prime $p$. If $n \ge 2$, nonvanishing is new for all $j \ge 2$. Consequently, the virtual cohomological dimension of $\widehatΓ$ 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 $Γ$ and $j < \frac{n+1}{2}$, restriction ${H^j(\widehatΓ, \mathbb{F}_p) \to H^j(Γ, \mathbb{F}_p)}$ 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 $n=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20847 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomological nonvanishing for algebraic fundamental groups of ball quotients Stover, Matthew Algebraic Geometry Geometric Topology Number Theory Suppose $Γ< \mathrm{PU}(n,1)$ is a cocompact arithmetic lattice of simplest type with profinite completion $\widehatΓ$. This paper proves there is an open subgroup $\widehatΓ_0 \le \widehatΓ$ such that $H^j(\widehatΔ, \mathbb{F}_p)$ is nontrivial for every open subgroup ${\widehatΔ \le \widehatΓ_0}$, $j \le 2n$, and sufficiently large prime $p$. If $n \ge 2$, nonvanishing is new for all $j \ge 2$. Consequently, the virtual cohomological dimension of $\widehatΓ$ 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 $Γ$ and $j < \frac{n+1}{2}$, restriction ${H^j(\widehatΓ, \mathbb{F}_p) \to H^j(Γ, \mathbb{F}_p)}$ 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 $n=1$. |
| title | Cohomological nonvanishing for algebraic fundamental groups of ball quotients |
| topic | Algebraic Geometry Geometric Topology Number Theory |
| url | https://arxiv.org/abs/2508.20847 |