Orders on free groups induced by oriented words

Abstract

For every finite rank k, k>1, we explicitly construct (2k)! left orders on the free group F_k of rank k. Each order is induced by a word of length 2k in which each generator of F_k and its inverse appear exactly once. For each of these (2k)! words we define a real valued function on F_k, which is shown to be a quasi-character with small relative defect and which is used as a weight function to define the corresponding order (the elements of F_k which evaluate to positive real numbers are declared positive in the group). Some of the orders we define on F_k are extensions of the usual lexicographic order on the positive monoid and some have word reversible positive cones. We characterize the defining words leading to orders of either of these two types.