A1 Refereed original research article in a scientific journal
On the growth of actions of free products
Authors: Le Boudec, Adrien; Matte Bon, Nicolás; Salo, Ville
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Publishing place: BERLIN
Publication year: 2025
Journal: Groups, Geometry, and Dynamics
Journal name in source: Groups, Geometry, and Dynamics
Journal acronym: GROUP GEOM DYNAM
Volume: 19
Issue: 2
First page : 661
Last page: 680
Number of pages: 20
ISSN: 1661-7207
eISSN: 1661-7215
DOI: https://doi.org/10.4171/GGD/893
Publication's open availability at the time of reporting: Open Access
Publication channel's open availability : Open Access publication channel
Web address : https://doi.org/10.4171/ggd/893
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/499750040
Self-archived copy's licence: CC BY
Self-archived copy's version: Publisher`s PDF
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 F(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.
Keywords:
group actions
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Funding information in the publication:
This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Universite de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency.
