Stability in the homology of unipotent groups

Let R be a (not necessarily commutative) ring whose additive group is finitely generated and let U-n (R) subset of GL(n) (R) be the group of upper-triangular unipotent matrices over R. We study how the homology groups of U-n (R) vary with n from the point of view of representation stability. Our main theorem asserts that if for each n we have representations M-n of U-n (R) over a ring k that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule left perpendicular n right perpendicular bar right arrow H-i(U-n(R), M-n) defines a finitely generated OI-module. As a consequence, if k is a field then dim Hi(U-n(R), k) is eventually equal to a polynomial in n. We also prove similar results for the Iwahori subgroups of GL(n)(O) for number rings O.