More on subgroups of R. Thompson group F


Together with Gili Golan we wrote another paper on F. Note that  Savchuk proved that for every number \alpha in (0, 1) the subgroup of all elements of F that fix \alpha is maximal (and its Schreier graph is amenable). He asked if F has any other maximal subgroups of infinite index. That problem is soved (in the negative) in our paper. In fact we give two solutions.

First we  show that Jones’ subgroup \vec F of F is maximal in a certain subgroup of index 2 of F which itself is isomorphic to F. (Recall that \vec F is isomorphic to the Thompson-Brown group F_3 and is 3-generated.) Moreover, we prove that there are exactly three subgroups of F containing \vec F.  The preimage of \vec F under the natural injective endomorphism of F is then a maximal subgroup of F and it is not difficult to show that this subgroup does not fix a point in the open unit interval. The fact that there are only three subgroups of F containing Jones’ subgroup is very counter-intuitive and answers a question by Saul Schleimer.

The second solution gives many implicit examples (which may or may not be finitely generated). Note that the generator x_0 of F does not fix any point in the open unit interval. Now take any element g such H=\langle x_0, g\rangle is a proper subgroup of F and not inside any proper subgroup of finite index. Then any maximal subgroup  containing H is an example we need. The fact that H is not inside any proper subgroup of finite index is easy to establish. Indeed, every non-Abelian homomorphic image of F is F itself. So it is enough to show that the images of x_0, g in F/[F,F] generate the whole F/[F,F]. For example, one can take g=x_1x_2x_1^{-1}.

The non-trivial thing is to prove that H is proper. Indeed, the only previously known (more precisely – previously published) way to show that a subgroup of F is proper is to show that it is inside some proper subgroup of finite index of F or fixes a point on the open unit interval. We employ a 2-dimensional analog of Stallings’ solution of the membership problem for subgroups of free groups. For every  subgroup of F viewed as the diagram group over the Dunce hat, we construct what we called a Stallings 2-core which is a directed 2-complex with a distinguished 1-path p and a map into the Dunce hat (which is also a directed 2-complex). In the case of the free group, the Stallings core is an automaton which accepts a reduced word if and only if it belongs to the subgroup. In the case of F, the Stallings 2-core is a 2-automaton which “accepts” reduced diagrams from H (but not only these diagrams). In the case of g=x_1x_2x_1^{-1} we construct the Stallings 2-core and show that x_1 is not “accepted” by it. So x_1\not\in H, and H is indeed a proper subgroup of F. Although the method of Stallings 2-cores was never published before, we discussed it with Victor Guba long ago (in 1998 or even earlier). We wanted to write a paper about it but never had time to do that.

Note that the question of when the Stallings 2-core of a subgroup H of a diagram group accepts only the diagrams from H is very interesting because in that case, H is a diagram group itself. This is how, together with Victor Guba, we represented the derived subgroup [F,F] as a diagram group (see Theorem 26 in this paper).

If U is a finite set of finite binary fractions, then it is easy to see that the subgroup H_U of all elements of F that fix each number from U is just the direct product of |U|+1 copies of F. Savchuk studied these subgroups too, proving that their Schreier graphs are also amenable. We prove in our paper that for every U there are only finitely many subgroups of F containing H_U (moreover the lattice of subgroups of F containing H_U is anti-isomorphic to the Boolean lattice of subsets of U). This implies that F is quasi-residually finite (in our terminology), that is it has a decreasing sequence of finitely generated subgroups F > P_1 > P_2 > ... such that the intersection of these subgroups is trivial and for each i there are only finitely many subgroups of F containing P_i.   That is also a counter-intuitive property of F.

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