Partial actions of groups on profinite spaces

Luis Martínez; Héctor Pinedo; Andrés Villamizar

doi:10.4995/agt.2024.18049

Authors

Luis Martínez Universidad Nacional Autónoma de México https://orcid.org/0000-0002-3957-3119
Héctor Pinedo Industrial University of Santander https://orcid.org/0000-0003-4432-419X
Andrés Villamizar University of Pamplona https://orcid.org/0009-0004-6458-644X

DOI:

https://doi.org/10.4995/agt.2024.18049

Keywords:

partial action, profinite space, Orbit equivalence relation, clopen sets, globalization, continuous section, reflective subcategory, enveloping action

Abstract

We show that for a partial action η with closed domain of a compact group G on a profinite space X the space of orbits X/~_G is profinite, this leads to the fact that when G is profinite the enveloping space X_G is also profinite. Moreover, we provide conditions for the induced quotient map π_G : X → X / ∼_G of η to have a continuous section. Relations between continuous sections of π_G and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

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Author Biographies

Luis Martínez, Universidad Nacional Autónoma de México

Departamento de Matemáticas, Facultad de Ciencias

Héctor Pinedo, Industrial University of Santander

Escuela de Matemáticas, Facultad de Ciencias

Andrés Villamizar, University of Pamplona

Departamento de Matemáticas, Facultad de Ciencias Básicas

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