Vol. 304, No. 1, 2020

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Commensurability growth of branch groups

Khalid Bou-Rabee, Rachel Skipper and Daniel Studenmund

Vol. 304 (2020), No. 1, 43–54

Fixing a subgroup Γ in a group G, the commensurability growth function assigns to each n the cardinality of the set of subgroups Δ of G with [Γ : Γ Δ][Δ : Γ Δ] = n. For pairs Γ A, where A is the automorphism group of a p-regular rooted tree and Γ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups Γ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta–Sidki groups, etc.) acting on p-regular rooted trees, this function is precisely 0 for any n = pk .

commensurators, branch groups, residually finite groups
Mathematical Subject Classification 2010
Primary: 20B07, 20E26
Secondary: 20K10
Received: 30 August 2018
Revised: 20 July 2019
Accepted: 20 July 2019
Published: 18 January 2020
Khalid Bou-Rabee
Department of Mathematics
The City College of New York
New York, NY
United States
Rachel Skipper
Department of Mathematics
Ohio State University
Columbus, OH
United States
Daniel Studenmund
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States