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Research Papers
On the stabilizers of finite sets of numbers in the R. Thompson group $F$
G. Golan,
M. Sapir Vanderbilt University, 2201 West End Ave, Nashville, TN 37235, USA
Abstract:
The subgroups
$H_U$ of the R. Thompson group
$F$ that are stabilizers of finite sets
$U$ of numbers in the interval
$(0,1)$ are studied. The algebraic structure of
$H_U$ is described and it is proved that the stabilizer
$H_U$ is finitely generated if and only if
$U$ consists of rational numbers. It is also shown that such subgroups are isomorphic surprisingly often. In particular, if finite sets
$U\subset[0,1]$ and
$V\subset[0,1]$ consist of rational numbers that are not finite binary fractions, and
$|U|=|V|$, then the stabilizers of
$U$ and
$V$ are isomorphic. In fact these subgroups are conjugate inside a subgroup
$\bar F<\operatorname{Homeo}([0,1])$ that is the completion of
$F$ with respect to what is called the Hamming metric on
$F$. Moreover the conjugator can be found in a certain subgroup
$\mathcal F<\bar F$ which consists of possibly infinite tree-diagrams with finitely many infinite branches. It is also shown that the group
$\mathcal F$ is non-amenable.
Keywords:
Thompson group $F$, stabilizers. Received: 15.05.2016
Language: English
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