Right-angled Artin subgroups of right-angled Coxeter and Artin groups

We determine when certain natural classes of subgroups of right-angled Coxeter groups (RACGs) and right-angled Artin group (RAAGs) are themselves RAAGs. We first consider subgroups of RACGs generated by elements that are products of two non-commuting RACG generators; these were introduced in LaForge's thesis \cite{laforge}. Within the class of $2$-dimensional RACGs, we give a complete characterization of when such a subgroup is a finite-index RAAG system. As an application, we show that any $2$-dimensional, one-ended RACG with planar defining graph is quasi-isometric to a RAAG if and only if it contains an index $4$ subgroup isomorphic to a RAAG. We also give applications to other families of RACGs whose defining graphs are not planar. Next, we show that every subgroup of a RAAG generated by conjugates of RAAG generators is itself a RAAG. This result is analogous to a classical result of Deodhar and Dyer for Coxeter groups. Our method of proof, unlike in the classical case, utilizes disk diagrams and is geometric in nature.