Given G, a fuchsian group and a finite sub set A of G. Does there exist a finite index subgroup H in G such that inter section of A with H is empty? 



finite index subgroup of a fuchsian group
The answer is yes. The group is residually finite. This means that if $A$ is taken to be a finite set of elements of the Fuchsian group $G$ $not \quad containing\quad identity$, then there exists a finite quotient of $G$ where no element of $A$ is trivial. The kernel of this quotient map is a finite index subgroup$H$ not intersecting $A$.


On the other hand, if $A$ contains identity, then of course the statement is false. 


