Partial actions of groups on profinite spaces




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


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 XG is also profinite. Moreover, we provide conditions for the induced quotient map πG  : X → X / ∼G  of η to have a continuous section. Relations between continuous sections of πG 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


H. H. Wicke and J. M. Worrell, Jr., Open continuous mappings of spaces having bases of countable order, Duke Math. J. 34 (1967), 255-271.

F. Abadie, Enveloping actions and Takai duality for partial actions. Journal of Funct. Anal. 197 (2003), 14-67.

J. Avila, S. Buitrago and S. Zapata, The category of partial actions of a group: some constructions, Int. J. Pure Appl. Math. 95, no. 1 (2014), 45-56.

M. Dokuchaev, Recent developments around partial actions, Sao Paulo J. Math. Sci. 13, no. 1 (2019), 195-247.

R. Exel, Circle actions on $C^*$-algebras, partial automorphisms and generalized Pimsner-Voiculescu exact sequences, J. Funct. Anal. 122, no. 3 (1994), 361-401.

R. Exel, Twisted partial actions: a classification of regular $C^*$-algebraic bundles, Proc. London Math. Soc. 74 (1997), 417-443.

R. Exel, Partial actions of group and actions of inverse semigroups, Proc. Am. Math. Soc. 126, no. 12 (1998), 3481-3494.

R. Exel, Partial dynamical systems, Fell bundles and applications, Mathematical surveys and monographs; volume 224, Providence, Rhode Island: American Mathematical Society, 2017.

R. Exel, T. Giordano, and D. Gonçalves, Enveloping algebras of partial actions as groupoid $C^*$-algebras, J. Operator Theory 65 (2011), 197-210.

J. Kellendonk and M. V. Lawson, Partial Actions of Groups, International Journal of Algebra and Computation 14 (2004), 87-114.

M. Khrypchenko and B. Novikov, Reflectors and globalizations of partial actions of groups, J. Aust. Math. Soc. 104, no. 3 (2018), 358-379.

K. McClanaham, K-theory for partial crossed products by discrete groups, J. Funct. Anal. 130 (1995), 77-117.

S. MacLane, Categories for the Working Mathematician, Springer, New York-Heidelberg-Berlin, 1971.

A. Magid, The separable Galois theory of commutative rings, second edition, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1974.

H. Pinedo and C. Uzcátegui, Polish globalization of Polish group partial actions, Math. Log. Quart. 63 6 (2017), 481-490.

H. Pinedo and C. Uzcátegui, Borel globalization of partial actions of Polish groups, Arch. Mat. Log. 57 (2018), 617-627.

J. C. Quigg and I. Raeburn, Characterizations of crossed products by partial actions, J. Operator Theory 37 (1997), 311-340.

L. Ribes and P. Zalesski, Profinite Groups, Springer, Berlin, 2000.

B. Steinberg, Partial actions of groups on cell complexes, Monatsh. Math. 138, no. 2 (2003), 159-170.




How to Cite

L. Martínez, H. Pinedo, and A. Villamizar, “Partial actions of groups on profinite spaces”, Appl. Gen. Topol., vol. 25, no. 1, pp. 143–157, Apr. 2024.



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