Amenable groups, mean topological dimension and subshifts
by
Fabrice Krieger
IRMA, Strasbourg

Given a group G, a G-space is a compact metrizable space endowed with a countinuous G-action. One says that a G-space is minimal if all its orbits are dense. A theorem of Jaworski says that if G is an abelian group, any minimal G-space X of finite topological dimension is embeddable in the G-shift on [0, 1]^G. In the case G=Z, E. Lindenstrauss and B. Weiss give a counterexample which shows that one cannot remove the finiteness hypothesis on the topological dimension in Jaworski's result. We will generalize this result and prove that if G is an infinite countable residually finite amenable group, there is a minimal G-space which is not embbedable in the G-shift on [0, 1]^G. In particular, one cannot suppress the finiteness hypothesis on the topological dimension of X in Jaworski's theorem if G is finitely generated. The essential tool used in the proof is "mean topological dimension", which is a topological invariant for amenable group actions, introduced by M. Gromov. By using this invariant, one reduces the problem to the construction of a minimal G-space of mean topological dimension strictly larger than that of the G-shift [0, 1]^G.

Date received: February 15, 2006