Riemannian groupoids and solitons for three-dimensional homogeneous ricci and cross-curvature flows

In this paper, we investigate the behavior of three-dimensional homogeneous solutions of the cross-curvature flow using Riemannian groupoids. The Riemannian groupoid technique, originally introduced by J. Lott, allows us to investigate the long-term behavior of collapsing solutions of the flow, producing solitons in the limit. We also review Lott's results on the long-term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross-curvature soliton metrics on Sol and Nil, and show that the cross-curvature flow of SL(2,R) limits to Sol.