Scale functions and tree ends (Q2726692)

The authors examine a class of totally disconnected groups consisting of partial direct products on an index set. For such a group, the authors find the scale function and characterise the tidy subgroups for automorphisms arising from permutations of the index set. When the index set is a finitely generated free group and the permutation is translation by an element \(x\) of the group, the authors show that the scale depends on a cyclically reduced form of \(x\) and the tidy subgroup on the element which conjugates \(x\) to its cyclically reduced form.