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 $k\cdot X+l\cdot X$, 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.