Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ 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 $\mathbf{ k}$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $[n] \mapsto \widetilde{ H}_i(U_n(R),M_n)$ defines a finitely generated OI-module. As a consequence, if $\mathbf{ k}$ is a field then $dim \widetilde{ H}_i(U_n(R),\mathbf{ k})$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of $GL_n(\mathcal{ O})$ for number rings $\mathcal{ O}$.
Categories: math.AT math.GR math.RT