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Abstract
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Fixing a subgroup
in a group
,
the commensurability growth function assigns to each
the cardinality of
the set of subgroups
of
with
. For pairs
, where
is the automorphism
group of a
-regular
rooted tree and
is finitely generated, we show that this function can take on finite,
countable, or uncountable cardinals. For almost all known branch groups
(the first Grigorchuk group, the twisted twin Grigorchuk
group, Pervova groups, Gupta–Sidki groups, etc.) acting on
-regular rooted trees,
this function is precisely
for any
.
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Keywords
commensurators, branch groups, residually finite groups
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Mathematical Subject Classification 2010
Primary: 20B07, 20E26
Secondary: 20K10
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Milestones
Received: 30 August 2018
Revised: 20 July 2019
Accepted: 20 July 2019
Published: 18 January 2020
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