Surfaces and other Peano Continua with no Generic Chains

The space of chains on a compact connected space encodes all the different ways of continuously growing out of a point until exhausting the space. A chain is generic if its orbit under the action of the underlying homeomorphism group is comeager: a space X has a generic chain if there is essentially just one type of chain on X. Gutman, Tsankov and Zucker proved that compact manifolds of dimension at least 3 do not have a generic chain. We extend and generalize their result, covering a large class of spaces which includes all compact surfaces except for the sphere and the real projective plane - for which the question remains open - as well as all other homogeneous Peano continua, circle excluded. If the spaces are moreover strongly locally homogeneous, which is the case for any closed manifold as well as the Menger curve, we prove that chains cannot be classified up to homeomorphism by countable structures, and that the underlying homomorphism groups have non-metrizable universal minimal flows. This is in contrast with the case of 1-dimensional manifolds: their homomorphism groups have metrizable universal minimal flow and we show that their chains are classifiable by countable structures and have a generic element. The proof of the main result relies on crafting a dictionary between open sets of chains on one side, and walks on finite connected graphs on the other. We then establish a novel combinatorial necessary condition for the existence of a generic chain, an off-by-one weak amalgamation principle, and prove that it is not satisfied under the hypotheses of our theorem.