The Cohomology of Lattices in SL(2, ℂ)
The Cohomology of Lattices in SL(2, ℂ)
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DOI：
10.1080/10586458.2010.10129067
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发表时间：
2008-08
影响因子：
0.5
通讯作者：
Tobias Finis;F. Grunewald;P. Tirao
中科院分区：
文献类型：
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作者：
Tobias Finis;F. Grunewald;P. Tirao
本文给出了上同调空间H1（Γ，En）的维数行为的理论结果和实验数据，其中Γ是SL（2，ε）中的格，n ∈ ε {0}是标准的自对偶模.对于虚二次数域上的整数环O，在Γ = SL（2，O）的情形下，我们使提升理论显式化，得到了n中线性的下界.我们提出了大量的实验数据，这种情况下，以及一些几何构造和大多数nonarithmetic组。SL（2，O）的计算使我们发现了两个上同调中具有非提升类的实例。在一般情况下，我们还得到了任意固定格Γ的长度为O（n2/logn）的上界.我们讨论了一些新的问题和建议，我们的结果和我们的实验数据。
This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H 1(Γ,E n ), where Γ is a lattice in SL(2,ℂ) and , n ∈ ℕ ∪ {0}, is one of the standard self-dual modules. In the case Γ = SL(2,O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We present a large amount of experimental data for this case, as well as for some geometrically constructed and mostly nonarithmetic groups. The computations for SL(2,O) lead us to discover two instances with nonlifted classes in the cohomology. We also derive an upper bound of size O(n 2/ log n) for any fixed lattice Γ in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.