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User:Antti Karttunen/Speculations/On the connection of A089840 with Thompsons groups
This is another page in my series of speculations. Here I am interested about the apparent connection of group A089840 with Thompson's groups F, T and V.
Problem
While it may first look that in the clause-representation of A089840 the clauses are just elements of Thompson's group , there is an important difference that in bijections like *A074679 and *A074680 the internal nodes (marked as "x"'s in ASCII graphics illustrations) are not labeled, and thus, those bijections don't specify anything about their relative order (between the argument tree and the result). But in contrast to A089840, Thompson's groups act on infinite binary trees, whose internal vertices are labeled and there are no external vertices, because the tree is infinite to all directions away from the root. The only way to distinguish one element's effect from other element's effect, is how the labelings of the internal vertices will change.
However, even although the other group (usually) acts on finite binary trees and the other on infinite ones, that does not need in itself to be a show-stopper. After all, we can construct an isomorphism in step-by-step fashion[1] between the automorphism group of infinite binary trees and the group of ordinary Catalan bijections that act on finite binary trees. See "psi-isomorphism" given in A153141.
Perhaps
Consider those Catalan bijections that fix any finite binary tree that is large enough. That is, for such bijections, there is always a minimal such binary tree, that will be fixed by the bijection and whose all super-trees will be fixed as well. Examples of such bijections are for example the identity bijection *A001477, which of course fixes everything, but also a bijection like *A154126, for which such a minimal tree is , that will be fixed, as well as any tree that is its super-tree. The set of these kind of bijections are closed with respect to the composition (see also User:Antti_Karttunen/Speculations/On internal construction of A089840), as well as taking of inverses. That is, they form a subgroup () of all Catalan (size-preserving) bijections (group ). Provided that the above mentioned problems can be brushed under the carpet, my conjecture is then that the quotient group () is isomorphic to some of the Thompson's groups (probably ), when the group is some suitably selected subgroup of the whole group . (Should be a countable group? See below.)
Note: the group is uncountable, while all Thompson's groups have a finite set of generators, and as long as we consider only elements which are finite products of those generators, they are countable groups. However, consider an order-preserving permutation of the positive rational numbers (whose Stern-Brocot-based signature permutation is given by A065249). If we apply Minkowski's question-mark function to it, i.e. we then have a function and apply yet some other transformation, then we should get an order-preserving permutation of rationals on range (??? Have forgotten too much...). Should we accept also these kind "limits of infinite products" as elements of Thompson's ? (some kind of an extended version, perhaps?)
Well, I think it's better to take the group as A089840, and forget for a while any recursively defined binary tree bijections.
Now, somebody should do all the hard (???) work, e.g. in checking that the subgroup is normal in A089840, and whether it always holds that the product for any two elements and of A089840, which have the same first clause.
References and notes
- ↑ Is it actually an instance of strong amalgamation in Fraïssé's model theory? See Age (model theory)—Wikipedia.org.