On subgroups of R. Thompson’s group 𝐹

Golan, Gili; Sapir, Mark

doi:10.1090/tran/6982

On subgroups of R. Thompson’s group $F$
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Trans. Amer. Math. Soc. 369 (2017), 8857-8878 Request permission

Abstract:

We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones’ subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings’ core graphs and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>\cdots$ such that $\cap H_i=\{1\}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.

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