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A geometric interpretation of the Schützenberger group of a minimal subshift

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Arkiv för Matematik

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 Berthé et al.

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Author information

Authors and Affiliations

  1. CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, PT-4169-007, Porto, Portugal

    Jorge Almeida

  2. CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, EC Santa Cruz, PT-3001-501, Coimbra, Portugal

    Alfredo Costa

Authors
  1. Jorge Almeida
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  2. Alfredo Costa
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Correspondence to Jorge Almeida.

Additional information

Work partially supported respectively by CMUP (UID/MAT/00144/2013) and CMUC (UID/MAT/00324/2013), which are funded by FCT (Portugal) with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020.

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Almeida, J., Costa, A. A geometric interpretation of the Schützenberger group of a minimal subshift. Ark Mat 54, 243–275 (2016). https://doi.org/10.1007/s11512-016-0233-7

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  • DOI: https://doi.org/10.1007/s11512-016-0233-7

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