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Abstract
For every countable residually finite group G and every totally disconnected metric compactification \overleftarrow{G} of G , we construct an irregular Toeplitz subshift X\subseteq \{0,1\}^{G} whose maximal equicontinuous factor is \overleftarrow{G} and such that X is a topo-isomorphic extension of \overleftarrow{G} . When the acting group G is amenable, our construction yields new examples of mean-equicontinuous dynamical systems.
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