On the stabilizers of finite sets of numbers in the R. Thompson group F

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 ⊂ [0, 1] and V ⊂ [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 F < 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 F < F which consists of possibly infinite tree-diagrams with finitely many infinite branches. It is also shown that the group F is non-amenable.