Right-angled Artin subgroups and free products in one-relator groups

We investigate criteria ensuring that a one-relator group G contains a right-angled Artin subgroup A(Γ), corresponding to a finite graph Γ. In particular, we prove that if the positive submonoid T(Γ), of A(Γ), embeds into G then so does all of A(Γ), unless Γ is totally disconnected. As by-products of our methods we obtain characterisations of one-relator groups that have property Pₙₐᵢ and that are C^*-simple.