On the long-time behavior of type-III Ricci flow solutions

We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton's compactness theorem that does not assume a lower injectivity radius bound, in terms of Riemannian groupoids. Using this, we show that the long-time behavior of type-III Ricci flow solutions is governed by the dynamics of an R^+ action on a compact space.