** **

**Abstract:**

About 30 years ago Ruth Charney proved that
the homology of the

general linear group stabilized. In the case of GL(n,Z) this means that

if M is any constant coefficient module and
r is fixed, H_r(GL(n,Z),M)

doesn't depend on n if n is sufficiently
large. Recently, Frank Calegari

and Akshay Venkatesh asked: how do the Hecke
operators act on the stable

homology?

In the first half of my talk I will review
group homology, Hecke

operators, and the connection with Galois
representations. In the second

half I will investigate how the Hecke
operators act on the stable homology

and show that the Hecke eigenvalues
correspond to Galois representations

which are isomorphic to direct sums of
powers of the cyclotomic character.