Let $\mathcal{S}\_g$ denote the fundamental group of an oriented surface of genus $g\ge 2$.


Does $\mathcal{S}\_g$ contain subgroups $A$ and $B$ of finite index such that $A\cap B = \lbrace e\rbrace$?



Intersections of subgroups of surface groups
As Misha says in a comment, for any group $G$ with subgroups $A,B$, we have


$|G:A\cap B|\leq |G:A||G:B|$


(exercise). In particular, if $G$ is infinite (as here) then $A\cap B$ is non-trivial.


