On the growth of actions of free products

: Le Boudec, Adrien; Matte Bon, Nicolás; Salo, Ville

Publisher: European Mathematical Society - EMS - Publishing House GmbH

: BERLIN

: 2025

: Groups, Geometry, and Dynamics

: GROUP GEOM DYNAM

: 19

: 2

: 661

: 680

: 20

: 1661-7207

: 1661-7215

DOI: https://doi.org/10.4171/GGD/893

: https://doi.org/10.4171/ggd/893

: 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.

10.4171-ggd-893.pdf

:
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.