A geometric interpretation of the Sch\"utzenberger group of a minimal subshift

TL;DR

This paper provides a geometric interpretation of the Schützenberger group associated with minimal subshifts, linking it to inverse limits of fundamental groups of Rauzy graphs and offering criteria for its freeness.

Contribution

It establishes a geometric perspective on the Schützenberger group of minimal subshifts by relating it to Rauzy graphs and their fundamental groups, extending previous algebraic descriptions.

Findings

The Schützenberger group is isomorphic to the inverse limit of the profinite completions of Rauzy graph fundamental groups.

A geometric criterion for the freeness of the profinite group is provided.

The approach connects algebraic and geometric methods in symbolic dynamics.

Abstract

The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for freeness of the profinite group of a minimal subshift based on the Return Theorem of Berth\'e et. al.