Stability in the homology of unipotent groups

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ 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 $\mathcal{O}$.

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