We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set.

I have been constructing a space at infinity for right-angled artin groups where the boundaries of quasiconvex hyperbolic subgroups will embed in a natural way; but to construct it, I've been forced to add all redundant generators of the form $b=a_1...a_k$, where all if the $a_i$ are distinct commuting generators in the standard generating set for the RAAG (they can also be unversed of generators). This is just adding in all 'diagonal' generators, and corresponds in free abelian groups to changing from an $L^1$ metric to an $L^{\infty}$ metric.

My question is, will this make some quasi convex subgroups no longer convex? In the $\mathbb{Z}^2$ case (generated by $a,b$), I think that the subgroup generated by $a$ is quasi convex before adding the generators $ab$, $ab^{-1}$, etc. So my final question is, will non-elementary hyperbolic quasi convex subgroups stay quasi convex after adding in these generators?

**Edit**: the easiest example of a non-elementary hyperbolic subgroup of a RAAG is the following: take the pentagon RAAG generated by $a,b,c,d,e$. Then the subgroup generated by $ab^{-1},bc^{-1},cd^{-1},de^{-1}$ is a two-holed torus group with standard relations.