In my talk, I will discuss some key ideas of the construction. Some of the ingredients, like traveling salesman groups, are interesting independently, and turn out to be useful in other areas as well, e.g. in the theory of amenable groups. Also, the properties in question do not imply any (strong) finiteness condition for groups and it seems that one can unite all of the above in one general statement; I will end the talk with some speculations.
where the equivalence relation is post-composition by conjugation in Mod(S).
I'll present some results about the structure of the set XS(G), for an arbitrary finitely presented group G. Morally, XS(G) being infinite implies that G admits a splitting as a graph of groups, and the structure of the set XS(G) can (often) be understood via studying splittings of G.