The cohomology of lattices in sl(2, c)

Tobias Finis; Fritz Grunewald; Paulo Tirao

doi:10.1080/10586458.2010.10129067

The cohomology of lattices in sl(2, c)

Tobias Finis, Fritz Grunewald, Paulo Tirao

Research output: Contribution to journal › Article › peer-review

23 Scopus citations

Abstract

This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H¹(Γ, En), where Γ is a lattice in SL(2,ℂ) and (Formula Presented), n ∈ ℕ ∪ {0}, is one of the standard selfdual 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(n²/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.

Original language	English
Pages (from-to)	29-63
Number of pages	35
Journal	Experimental Mathematics
Volume	19
Issue number	1
DOIs	https://doi.org/10.1080/10586458.2010.10129067
State	Published - 2010
Externally published	Yes

Keywords

Automorphic forms
Cohomology of arithmetic groups
Kleinian groups

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10.1080/10586458.2010.10129067

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title = "The cohomology of lattices in sl(2, c)",

abstract = "This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H1(Γ, En), where Γ is a lattice in SL(2,ℂ) and (Formula Presented), n ∈ ℕ ∪ {0}, is one of the standard selfdual 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(n2/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.",

keywords = "Automorphic forms, Cohomology of arithmetic groups, Kleinian groups",

author = "Tobias Finis and Fritz Grunewald and Paulo Tirao",

note = "Funding Information: The second author was supported by the DFG-Graduiertenkolleg 1150 (Homotopy and Cohomology) and the DFG-Forschergruppe 790 (Classification of Algebraic Surfaces and Compact Complex Manifolds).",

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doi = "10.1080/10586458.2010.10129067",

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T1 - The cohomology of lattices in sl(2, c)

AU - Finis, Tobias

AU - Grunewald, Fritz

AU - Tirao, Paulo

N1 - Funding Information: The second author was supported by the DFG-Graduiertenkolleg 1150 (Homotopy and Cohomology) and the DFG-Forschergruppe 790 (Classification of Algebraic Surfaces and Compact Complex Manifolds).

PY - 2010

Y1 - 2010

N2 - This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H1(Γ, En), where Γ is a lattice in SL(2,ℂ) and (Formula Presented), n ∈ ℕ ∪ {0}, is one of the standard selfdual 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(n2/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.

AB - This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H1(Γ, En), where Γ is a lattice in SL(2,ℂ) and (Formula Presented), n ∈ ℕ ∪ {0}, is one of the standard selfdual 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(n2/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.

KW - Automorphic forms

KW - Cohomology of arithmetic groups

KW - Kleinian groups

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DO - 10.1080/10586458.2010.10129067

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SN - 1058-6458

VL - 19

SP - 29

EP - 63

JO - Experimental Mathematics

JF - Experimental Mathematics

IS - 1

ER -

The cohomology of lattices in sl(2, c)

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Keywords

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