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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 $F>H_1>H_2>...$ such that $cap H_i={1}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.
We prove that Thompsons group $F$ has a subgroup $H$ such that the conjugacy problem in $H$ is undecidable and the membership problem in $H$ is easily decidable. The subgroup $H$ of $F$ is a closed subgroup of $F$. That is, every function in $F$ whic
We produce a sequence of markings $S_k$ of Thompsons group $F$ within the space ${mathcal G}_n$ of all marked $n$-generator groups so that the sequence $(F,S_k)$ converges to the free group on $n$ generators, for $n geq 3$. In addition, we give prese
We prove that under two natural probabilistic models (studied by Cleary, Elder, Rechnitzer and Taback), the probability that a random pair of elements of Thompsons group $F$ generate the entire group is positive. We also prove that for any $k$-genera
The definition of graph automatic groups by Kharlampovich, Khoussainov and Miasnikov and its extension to C-graph automatic by Murray Elder and the first author raise the question of whether Thompsons group F is graph automatic. We define a language
It is not known whether Thompsons group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to C-graph automatic by the authors, a compelling question