where the equivalence relation is post-composition by conjugation in Mod(S).
I'll present some results about the structure of the set X_{S}(G), for an arbitrary finitely presented group G. Morally, X_{S}(G) being infinite implies that G admits a splitting as a graph of groups, and the structure of the set X_{S}(G) can (often) be understood via studying splittings of G.
In this lecture, using the moving half-space method and the one-sided gradient flow of distance functions on Alexandrov spaces, we establish an extension of Perelman's soul theorem for singular spaces:
“Let M be a non-compact, complete, n-dimensional Alexandrov space of non-negative curvature. Suppose that M has no boundary and M has positive curvature on a non-empty open subset. Then M must be contractible".
This is a joint work with B. Dai and J. Mei.