Coxeter subgroups of Coxeter groups vs Artin subgroups of Artin groups

Question

Given two Coxeter groups $W(\Gamma)$ and $W(\gamma)$ of equal rank and their Artin groups $A(\Gamma)$ and $A(\gamma)$, is $W(\Gamma)\supset W(\gamma)$ (as reflection groups) equivalent to $A(\Gamma)\supset A(\gamma)$ (as abstract groups)?

Answer 1

No, there are counterexamples already in the right-angled case.

Let $C_n$ denote the cycle of length $n \geq 5$ and let $C_n^\ast$ denote the graph obtained by gluing two distinct copies of $C_n$ along a path of length two. Observe that, in $C_n^\ast$, there exists induced tree $T_n$ that has $n$ vertices but that is not a line. On the one hand:

Proposition. A Coxeter group is isomorphic to a subgroup of the right-angled Coxeter group $C(C_n)$ if and only if it is isomorphic to a right-angled Coxeter group $C(\Gamma)$ where $\Gamma$ is either a disjoint union of segment or a single cycle of length divisible by $n-4$.

See Proposition 4.21 in the article Morphisms between right-angled Coxeter groups and the embedding problem in dimension two. As a particular case, the right-angled Coxeter group $C(T_n)$ does not embed (abstractly) in $C(C_n)$. (Observe that, by construction, the two Coxeter groups have equal ranks.)

On the other hand, the right-angled Artin group $A(C_n^\ast)$ (and a fortiori $A(T_n)$) is isomorphic to a subgroup of $A(C_n)$. See for instance Corollary 5 in Embedability between right-angled Artin groups.

It is possible to construct many other examples. For instance by using the fact that, if $\Gamma$ has diameter $\geq 3$, then $A(F)$ embeds in $A(\Gamma)$ for every finite forest $F$. On the other hand, given two finite trees $R$ and $S$, $C(R)$ is isomorphic to a subgroup of $C(S)$ if and only if there exists a graph morphism $\varphi : R → S$ that sends a vertex of degree $2$ to a vertex of degree $\geq 2$ and a vertex of degree $\geq 3$ to a vertex of degree $\geq 3$.