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


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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. 

