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

Bui Anh Tuan (1) , Alexander D. Rahm (2, 3) , Matthias Wendt (4)

1 VNU-HCM - Vietnam National University, Ho Chi Minh City / Đại học Quốc gia TP. Hồ Chí Minh
2 GAATI - Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information
3 UPF - Université de la Polynésie Française
4 Bergische Universität Wuppertal

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.

Domaines: K-théorie et homologie [math.KT]

Bui Anh Tuan, Alexander D. Rahm, Matthias Wendt. On Farrell-Tate cohomology of GL(3) over rings of quadratic integers. 2022. ⟨hal-02435963v2⟩