  Let $W$ be a right-angled Coxeter group corresponding to a finite
non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem
says that $\mathcal{G}^c$ is connected if and only if for any infinite index
quasiconvex subgroup $H$ of $W$ and any finite subset $\{ \gamma_1, \ldots ,
\gamma_n \} \subset W \setminus H$ there is a surjection $f$ from $W$ to a
finite alternating group such that $f (\gamma_i) \notin f (H)$. A corollary is
that a right-angled Artin group splits as a direct product of cyclic groups and
groups with many alternating quotients in the above sense.
  Similarly, finitely generated subgroups of closed, orientable, hyperbolic
surface groups can be separated from finitely many elements in an alternating
quotient, answering positively a conjecture of Wilton.
