Given a finite graph $\Gamma$, one has the right-angled Artin group $A(\Gamma )$. Its generators $s_1, \dots s_n$ bijectively correspond to vertices of $\Gamma$ and the relators are $s_is_j=s_js_i$ provided the corresponding vertices are joined by an edge.

Let $A_i(\Gamma)$ be the group obtained from $A(\Gamma)$ by setting $s_i=1$; this corresponds to removing the vertex $i$ from $\Gamma$.

I know very little of these matters but it seems plausible that any nontrivial element of $A(\Gamma)$ projects to a nontrivial element of $A_i(\Gamma)$ for some $i$; is this correct?