$\DeclareMathOperator\Mod{Mod}$Let $S$ be a closed surface and $\Mod(S)$ be its mapping class group.


It is a well known fact, proved in the Primer on Mapping class groups for example, that the subgroup of $\Mod(S)$ generated by two Dehn twists $T\_a$ and $T\_b$ depends only on the geometric intersection number $i(a,b)$.


Indeed, if $i(a,b) \geq 2$ a ping pong lemma argument on the curve graph shows that $\langle T\_a, T\_b \rangle$ is a free group. Also, if $i(a,b) = 0$, then $\langle T\_a, T\_b \rangle$ is clearly free abelian.
Lastly, if $i(a,b) = 1$, it can be shown that $\langle T\_a, T\_b \rangle$ is a braid group on 3 strands, unless $S$ is a torus.


I am wondering whether such a statement can be proven in general, even without knowing a full classification of the groups that can appear. More precisely, if $\{a\_1, \cdots, a\_n\}$ and $\{b\_1, \cdots, b\_n\}$ are two collections of curves such that $i(a\_i,a\_j) = i(b\_i,b\_j)$ for all $i,j$, is it true that $\langle T\_{a\_1}, \cdots, T\_{a\_n} \rangle \simeq \langle T\_{b\_1}, \cdots, T\_{b\_n} \rangle$ as subgroups of $\Mod(S)$?.


It seems that the group $\langle T\_{a\_1}, \cdots, T\_{a\_n} \rangle$ depends only on a regular neighborhood $N\_a$ of the curves $a\_i$, which is then homeomorphic to any regular neighborhood $N\_b$ of the curves $b\_i$. But in general the injection $i \colon N\_a \to S$ does not induce an injective map on the mapping class group levels, so this does not seem to give a proper argument.


Any help is greatly appreciated!



Subgroups of Mod(S) generated by Dehn twists depend only on intersection numbers?
This is not true and I don't see the "right" side-conditions to make it true. Here is an example.


Suppose that $a\_1, a\_2, a\_3$ all lie in a single handle (surface of genus one, with one boundary component). and all meet exactly once, pairwise. Then the twist about $a\_3$ lies in the group generated by the others, giving the usual three-strand braid group.


Suppose that $b\_1, b\_2, b\_3$ all meet exactly once, pairwise. Let $N$ be a regular neighbourhood of their union. Assume that $N$ has genus one and three boundary components. Assume that all boundary components are essential in the ambient surface $S$. Then the group generated is some Artin group with three generators.


