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
Web address : https://doi.org/10.4171/ggd/893
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/499750040
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.
Downloadable publication This is an electronic reprint of the original article. |  |
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.
