Decidability and symbolic dynamics over groups

Abstract: 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.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic

Vienna Geometry and Analysis on Groups Seminar