Together with Gili Golan we wrote another paper on . Note that Savchuk proved that for every number in the subgroup of all elements of that fix is maximal (and its Schreier graph is amenable). He asked if 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 of is maximal in a certain subgroup of index of which itself is isomorphic to . (Recall that is isomorphic to the Thompson-Brown group and is 3-generated.) Moreover, we prove that there are exactly three subgroups of containing . The preimage of under the natural injective endomorphism of is then a maximal subgroup of 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 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 of does not fix any point in the open unit interval. Now take any element such is a proper subgroup of and not inside any proper subgroup of finite index. Then any maximal subgroup containing is an example we need. The fact that is not inside any proper subgroup of finite index is easy to establish. Indeed, every non-Abelian homomorphic image of is itself. So it is enough to show that the images of in generate the whole . For example, one can take .
The non-trivial thing is to prove that is proper. Indeed, the only previously known (more precisely – previously published) way to show that a subgroup of is proper is to show that it is inside some proper subgroup of finite index of 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 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 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 , the Stallings 2-core is a 2-automaton which “accepts” reduced diagrams from (but not only these diagrams). In the case of we construct the Stallings 2-core and show that is not “accepted” by it. So , and is indeed a proper subgroup of . 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 of a diagram group accepts only the diagrams from is very interesting because in that case, is a diagram group itself. This is how, together with Victor Guba, we represented the derived subgroup as a diagram group (see Theorem 26 in this paper).
If is a finite set of finite binary fractions, then it is easy to see that the subgroup of all elements of that fix each number from is just the direct product of copies of . Savchuk studied these subgroups too, proving that their Schreier graphs are also amenable. We prove in our paper that for every there are only finitely many subgroups of containing (moreover the lattice of subgroups of containing is anti-isomorphic to the Boolean lattice of subsets of ). This implies that is quasi-residually finite (in our terminology), that is it has a decreasing sequence of finitely generated subgroups such that the intersection of these subgroups is trivial and for each there are only finitely many subgroups of containing . That is also a counter-intuitive property of .