1École Normale Supérieure de Lyon, CNRS, France 2Université Claude Bernard Lyon 1, CNRS, Villeurbanne, France 3University of Turku, Turun yliopisto, Finland
Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 661-680
If G is a finitely generated group and X a G-set, the growth of the action of G on X is the function that measures the largest cardinality of a ball of radius n in the (possibly non-connected) Schreier graph Γ(G,X). We consider the following stability problem: if G,H are finitely generated groups admitting a faithful action of growth bounded above by a function f, does the free product G∗H also admit a faithful action of growth bounded above by f? We show that the answer is positive under additional assumptions, and negative in general. In the negative direction, our counter-examples are obtained with G either the commutator subgroup of the topological full group of a minimal and expansive homeomorphism of the Cantor space, or G a Houghton group. In both cases, the group G admits a faithful action of linear growth, and we show that G∗H admits no faithful action of subquadratic growth provided H is non-trivial. In the positive direction, we describe a class of groups that admit actions of linear growth and is closed under free products and exhibit examples within this class, among which the Grigorchuk group.
Adrien Le Boudec
1
;
Nicolás Matte Bon
2
;
Ville Salo
3
1
École Normale Supérieure de Lyon, CNRS, France
2
Université Claude Bernard Lyon 1, CNRS, Villeurbanne, France
3
University of Turku, Turun yliopisto, Finland
Adrien Le Boudec; Nicolás Matte Bon; Ville Salo. On the growth of actions of free products. Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 661-680. doi: 10.4171/ggd/893
@article{10_4171_ggd_893,
author = {Adrien Le Boudec and Nicol\'as Matte Bon and Ville Salo},
title = {On the growth of actions of free products},
journal = {Groups, geometry, and dynamics},
pages = {661--680},
year = {2025},
volume = {19},
number = {2},
doi = {10.4171/ggd/893},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/893/}
}
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AU - Nicolás Matte Bon
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