Borel proved that in low dimensions, the cohomology of a locally symmetric space can be represented not just by harmonic forms but by invariant forms. This implies that the k-th rational cohomology of SL_n(Z) is independent of n in a linear range n>= c*k , and tells us exactly what this "stable cohomology" is. In contrast, very little is known about the unstable cohomology, in higher dimensions outside this range.
In this talk I will explain a conjecture on a new kind of stability in the unstable cohomology of arithmetic groups like SL_n(Z). These conjectures deal with the "codimension-k" cohomology near the top dimension (the virtual cohomological dimension), and for SL_n(Z) they imply the cohomology vanishes there. Although the full conjecture is still open, I will explain how we proved it for codimension-0 and codimension-1. The key ingredient is a version of Poincare duality for these groups based on the algebra of modular symbols, and a new presentation for modular symbols. Joint work with Benson Farb and Andrew Putman.