Gianluca Basso "Surfaces and other Peano continua with no generic chains"
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Surfaces and other Peano continua with no generic chains
Abstract. A compact connected space has a generic chain if, roughly speaking,
there is essentially a unique way of covering the space by growing
continuously out of a point. The circle has a generic chain, but Gutman,
Tsankov and Zucker showed that no closed manifold of dimension at least
3 has a generic chain. We generalize and extend their result to a very
large class of spaces, comprising all compact surfaces except the sphere
and the real projective plane, as well as one dimensional curves like
the Menger sponge and Sierpinski carpet. Using classic results in
continuum theory, we reduce the problem to a purely
combinatorial statement about walks on finite connected graphs. The
theorem has dynamical consequences which can be interpreted as
non-rigidity results for the homeomorphism groups of the spaces
involved. This is joint work with Alessandro Codenotti and Andrea Vaccaro.