Stability in the homology of unipotent groups

Abstract

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n\left(R\right)\subset {GL}_n\left(R\right)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n\left(R\right)$ 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 $U_n\left(R\right)$ over a ring $k$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $\left[n\right]\mapsto H_i\left(U_n\left(R\right),M_n\right)$ defines a finitely generated $OI$-module. As a consequence, if $k$ is a field then $dim{H}_i\left(U_n\left(R\right),k\right)$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of ${GL}_n\left(\mathcal{𝒪}\right)$ for number rings $\mathcal{𝒪}$.