Self-simulable groups

Abstract: A configuration is a colouring of a finitely generated group by a finite alphabet. A subshift of finite type is a set of configurations defined by a finite collection of forbidden patterns. Subshifts of finite type naturally arise in the study of tilings and play are of great interest from a computational point of view and symbolic dynamics.

In the first part of the talk, we will address several questions that are classical in the case Z^2, but which lead to new and largely unexplored phenomena for general finitely generated groups. These include: - existence of a subshift of finite type containing at most one element of the alphabet. - existence of a subshift of finite type containing only aperiodic configurations (local rules force global behaviour). - decidability of the emptiness problem for subshift of finite type, given the set of forbidden patterns as input.

In the second part of the talk, we introduce a new class of groups. A finitely generated group is said to be self-simulable if every computable action of the group on an effectively closed zero-dimensional space is a topological factor of a subshift of finite type over that group. In other words, any “reasonable” group action can be encoded by local rules. We will show that such groups do exist, and that the class of self-simulable groups is stable under commensurability and under quasi-isometries among finitely presented groups. Finally, we will present several examples of self-simulable groups, including Thompson’s group V and higher-dimensional general linear groups.

This is a joint work with Sebastian Barbieri and Ville Salo.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic

Vienna Geometry and Analysis on Groups Seminar