Minimal length elements of Thompson's group \(F\)
This paper faces an important issue about finite presentations of Thompson's group \(F\): the problem of determining the length and the form of the minimal elements of \(F\). In particular, the author focuses on the two generator presentation \[ \langle a, b : [a b^{-1}, a^{-1} b a],\;[a b^{-1}, a^{-2} b a^2] \rangle. \] The main contribution is an effective algorithm for the computation of the length of an element in the group, with respect to the generators \(a\) and \(b\). The author also shows that it is possible to compute all minimal elements representing an element of \(F\). These results are based on the representation of elements by pairs of binary trees. In particular, the time complexity of both of the above algorithms is linear in the number of the vertices of the trees used to represent elements of \(F\). Though the proofs are quite involved, lemmas and theorems are well presented. Firstly, the nodes of any pair of trees representing an element of the group are classified and weighted, according to the generators associated with them. Then, it is shown that the sum of weights in the pair of trees is equal to the minimal length of the element with respect to the considered generators. I found this paper interesting and nice to read. Even if it is quite long (a bit more than 40 pages), the first 13 pages are a very useful introduction to Thompson's group \(F\) and its representations. In particular (more or less) known properties of tree diagrams and caret trees are reported in detail. My only small concern is about the conclusion of the work: the last algorithm for the computation of minimal elements of \(F\) deserves some further discussions and observations.
scientific article; zbMATH DE number 1977893