The Farrell–Tate and Bredon homology for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="normal">PSL</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> via cell subdivisions

Bui, Anh Tuan; Rahm, Alexander; Wendt, Matthias

doi:10.1016/j.jpaa.2018.10.002

Cited by 4 publications

(1 citation statement)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For machine calculations of Farrell-Tate or Bredon (co)homology, one needs cell complexes where cell stabilizers fix their cells pointwise. Bui, Wendt and the author have provided two algorithms computing an efficient subdivision of a complex to achieve this rigidity property [33]. Applying these algorithms to available cell complexes for PSL 4 (Z), they have computed the 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.…”

Section: Theorem 1 ([35]mentioning

confidence: 99%

An introduction to torsion subcomplex reduction

Rahm¹

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Theorem 1 ([35]mentioning

confidence: 99%

An introduction to torsion subcomplex reduction

Rahm¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

2019

Self Cite

View full text Add to dashboard Cite

We establish a dimension formula for the mod 2 cohomology of finite index subgroups in the Bianchi groups (SL 2 groups over the ring of integers in an imaginary quadratic number field). For congruence subgroups in the Bianchi groups, we provide an algorithm with which the parameters in our formula can be explicitly computed. We prove our formula with an analysis of the equivariant spectral sequence, combined with torsion subcomplex reduction.arXiv:1707.06078v2 [math.KT]

show abstract