On Farrell–Tate cohomology of SL2 over S-integers

Rahm, Alexander; Wendt, Matthias

doi:10.1016/j.jalgebra.2018.06.031

Cited by 4 publications

(5 citation statements)

References 23 publications

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“…The SL 2 groups over arbitrary number rings. Matthias Wendt and the author have established a complete description of the Farrell-Tate cohomology with odd torsion coefficients for all groups SL 2 (O K,S ), where O K,S is the ring of S-integers in an arbitrary number field K at an arbitrary non-empty finite set S of places of K containing the infinite places [35], based on an explicit description of conjugacy classes of finite cyclic subgroups and their normalizers in SL 2 (O K,S ).…”

Section: The Coxeter Groupsmentioning

confidence: 99%

An introduction to torsion subcomplex reduction

Rahm¹

2021

Preprint

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This survey paper introduces to a technique called Torsion Subcomplex Reduction (TSR) for computing torsion in the cohomology of discrete groups acting on suitable cell complexes. TSR enables one to skip machine computations on cell complexes, and to access directly the reduced torsion subcomplexes, which yields results on the cohomology of matrix groups in terms of formulas. TSR has already yielded general formulas for the cohomology of the tetrahedral Coxeter groups as well as, at odd torsion, of SL2 groups over arbitrary number rings. The latter formulas have allowed to refine the Quillen conjecture. Furthermore, progress has been made to adapt TSR to Bredon homology computations. In particular for the Bianchi groups, yielding their equivariant K-homology, and, by the Baum-Connes assembly map, the K-theory of their reduced C * -algebras. As a side application, TSR has allowed to provide dimension formulas for the Chen-Ruan orbifold cohomology of the complexified Bianchi orbifolds, and to prove Ruan's crepant resolution conjecture for all complexified Bianchi orbifolds.

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Section: The Coxeter Groupsmentioning

confidence: 99%

An introduction to torsion subcomplex reduction

Rahm¹

2021

Preprint

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“…As in [15,Section 5], the normalizer is an extension of the centralizer by an action of the stabilizer of the corresponding oriented module in the Galois group. We noted above that the Galois group Z/4Z exchanges the two orientations of the trivial module, hence the stabilizer is the subgroup Z/2Z ⊂ Z/4Z.…”

Section: Homological 5-torsion In Psl 4 (Z)mentioning

confidence: 99%

“…Applying the formulas from [15,Section 3], the Farrell-Tate cohomology of the normalizer is of the form F 5 [a ±2 2 ](b 3 1 ) ⊕2 ⊕ F 5 [a ±2 2 ](b 3 1 ) ⊕2 −1 where the lower subscript −1 indicates a degree shift by −1. The Hilbert-Poincaré series for the positive degrees is T 3 +2T 4 +T 5 1−T 4 = T 3 (1+T ) 2 1−T 4 .…”

Section: Homological 5-torsion In Psl 4 (Z)mentioning

confidence: 99%

“…Since the center of GL 4 (Z) is of order 2, the same is true for PGL 4 (Z). Now there is a necessary modification to deal with the case SL 4 (Z), along the lines of the discussion in [15]. While conjugacy classes of C 5 -subgroups in GL 4 (Z) correspond to isomorphism classes of Z[C 5 ]-modules, the conjugacy classes of C 5 -subgroups in SL 4 (Z) correspond to such modules equipped with an additional orientation, i.e., a choice of isomorphism det M ∼ = The centralizer of this C 5 -subgroup is the group of norm-1 units of Z[ζ 5 ], which by Dirichlet's unit theorem is isomorphic to ker…”

Section: Homological 5-torsion In Psl 4 (Z)mentioning

confidence: 99%

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The Farrell–Tate and Bredon homology for PSL4(Z) via cell subdivisions

Bui

Rahm

Wendt

2019

Journal of Pure and Applied Algebra

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We provide some new computations of Farrell-Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell-Tate or Bredon homology, one needs cell complexes where cell stabilizers fix their cells pointwise. We provide two algorithms computing an efficient subdivision of a complex to achieve this rigidity property. Applying these algorithms to available cell complexes for PSL 4 (Z) provides computations of Farrell-Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring.

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“…In effort of making a refinement of Quillens conjecture, Rahm and Wend, in their paper 11 , state that the conjecture is true for Farrel-Tate cohomology in the following cases.…”

Section: Quillen Conjecture For Farrell-tate Cohomologymentioning

confidence: 99%

A brief introduction to Quillen conjecture

Bui

Nguyen

2019

Sci. Tech. Dev. J.

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Introduction: In 1971, Quillen stated a conjecture that on cohomology of arithmetic groups, a certain module structure over the Chern classes of the containing general linear group is free. Over time, many efforts has been dedicated into this conjecture. Some verified its correctness, some disproved it. So, the original Quillens conjecture is not correct. However, this conjecture still has great impacts on the field cohomology of group, especially on cohomology of arithmetic groups. This paper is meant to give a brief survey on Quillen conjecture and finally present a recent result that this conjecture has been verified by the authors. Method: In this work, we investigate the key reasons that makes Quillen conjecture fails. We review two of the reasons: 1) the injectivity of the restriction map; 2) the non-free of the image of the Quillen homomorphism. Those two reasons play important roles in the study of the correctness of Quillen conjecture. Results: In section 4, we present the cohomology of ring which is isomorphic to the free module over . This confirms the Quillen conjecture. Conclusion: The scope of the conjecture is not correct in Quillens original statement. It has been disproved in many examples and also been proved in many cases. Then determining the conjectures correct range of validity still in need. The result in section 4 is one of the confirmation of the validity of the conjecture.

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On Farrell–Tate cohomology of SL2 over S-integers

Cited by 4 publications

References 23 publications

An introduction to torsion subcomplex reduction

An introduction to torsion subcomplex reduction

The Farrell–Tate and Bredon homology for PSL4(Z) via cell subdivisions

A brief introduction to Quillen conjecture

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