arXiv subjects

Bui Anh Tuan

Publications and source records attributed to Bui Anh Tuan.

2 recordsLinked to original sources

arXiv2022

On Farrell-Tate cohomology of GL(3) over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on Farrell-Tate cohomology for arithmetic groups. The conjugacy classification of cyclic subgroups is reduced to the classification of modules of group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell-Tate cohomology computations, we discuss in detail the homological torsion in PGL(3) over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.

math.KT↗

arXiv2019

Verification of the Quillen conjecture in the rank 2 imaginary quadratic case

We confirm a conjecture of Quillen in the case of the mod $2$ cohomology of arithmetic groups ${\rm SL}_2({\mathcal{O}}_{\mathbb{Q}(\sqrt{-m})}[\frac{1}{2}]\thinspace)$, where ${\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\thinspace)}$ is an imaginary quadratic ring of integers. To make explicit the free module structure on the cohomology ring conjectured by Quillen, we compute the mod $2$ cohomology of ${\rm SL}_2({\mathbb{Z}}[\sqrt{-2}\thinspace][\frac{1}{2}])$ via the amalgamated decomposition of the latter group.

math.AT↗