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 convex-cocompact 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 the conjecture of Wilton [9].

令 $W$ 是一个直角 Coxeter 群，对应于具有至少 $3$ 个顶点的有限非离散图 $\mathcal{G}$。我们的主要定理说 $\mathcal{G}^c$ 是连通的当且仅当对于 $W$ 的任何无限索引凸余紧子群 $H$ 和任何有限子集 $\{ \gamma_1, \ldots , \gamma_n \} \subset W \setminus H$ 存在从 $W$ 到有限交替群的投影 $f$，使得 $f (\gamma_i) \notin f (H)$。一个推论是，直角 Artin 群分裂为环状群和上述意义上具有许多交替商的群的直接乘积。

类似地，封闭的、可定向的、双曲曲面群的有限生成子群可以从交替商中的有限多个元素中分离出来，从而肯定地回答了威尔顿的猜想 [9]。