Hecke Operators on Stable Cohomology

TL;DR

This paper demonstrates the stabilization of completed cohomology groups for SL_N(Z) as N increases and shows that Hecke operators act trivially on this stable cohomology, revealing structural properties of these groups.

Contribution

It establishes the stabilization of cohomology groups and the triviality of Hecke operator action in the stable range, advancing understanding of automorphic forms and arithmetic groups.

Findings

Cohomology groups stabilize as N increases.

Hecke operators act trivially on stable cohomology.

Provides new insights into the structure of arithmetic groups.

Abstract

We prove that the completed cohomology groups of SL_N(Z) in fixed degree stabilize as N goes to infinity. We also prove that the action of Hecke operators on stable cohomology is trivial, in a precisely defined sense.