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 $A\cap B = \lbrace e\rbrace$?
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 $A\cap B = \lbrace e\rbrace$? 

closed as too localized by Benjamin Steinberg, Derek Holt, Misha, Alain Valette, Ian Agol Mar 13 '13 at 19:56This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question. 


As Misha says in a comment, for any group $G$ with subgroups $A,B$, we have $G:A\cap B\leq G:AG:B$ (exercise). In particular, if $G$ is infinite (as here) then $A\cap B$ is nontrivial. 

