Sums of Dilates in Ordered Groups

We address the ‘sums of dilates’ problem by looking for non-trivial lower bounds on sumsets of the form k·X + l·X, where k and l are non-zero integers and X is a subset of a possibly non-abelian group G (written additively). In particular, we investigate the extension of some results so far known only for the integers to the context of torsion-free or linearly orderable groups, either abelian or not.