Shifts of finite type are sets of biinfinite words (sequences of colors from a finite alphabet indexed in \(\mathbb{Z}\)) that avoid a finite collection of finite patterns. Their dynamical properties are very well understood thanks to their representation by matrices or finite graphs. When changing \(\mathbb{Z}\) into \(\mathbb{Z}^2\), the definition stays coherent, but most classical dynamical properties or invariants become intractable; one way to understand this is to consider this object as a computational model, capable of some algorithmic behavior.
Now, when changing \(\mathbb{Z}^2\) into any finitely generated group, it is not completely clear when the behavior is close to that of \(\mathbb{Z}\) or to that of \(\mathbb{Z}^2\). I will try to give some intuition on this open problem, survey what is known, and sketch some ideas that could help approach a solution.
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Passcode: A group is called an ________ group if it admits an invariant mean. (8 letters, lowercase)