On the geometric construction of cohomology classes for cocompact discrete subgroups of SL$_n(\mathbb{R})$ and SL$_n(\mathbb{C})$

Susanne Schimpf

Vol. 282 (2016), No. 2, 445–477

DOI: 10.2140/pjm.2016.282.445

Abstract

We construct nontrivial cohomology classes for certain cocompact discrete subgroups of ${SL}_n\left(ℝ\right)$ and ${SL}_n\left(ℂ\right)$ using a geometric method. The discrete subgroups are of arithmetic nature, i.e., they arise from arithmetic subgroups of suitably chosen algebraic groups. In certain cases, we show the nonvanishing of automorphic representations as a consequence.

Keywords

cohomology, arithmetic subgroup, special linear group, geometric cycle, automorphic representation

Mathematical Subject Classification 2010

Primary: 11F75

Secondary: 11F70, 11E57

Milestones

Received: 1 December 2014

Revised: 4 October 2015

Accepted: 6 October 2015

Published: 3 March 2016