Let $(M,hyp)$ be a closed hyperbolic surface. fix a point $m$ in $M$ and denote by $G=\pi\_1(M,m) $.
now let $\alpha$ and $\beta$ in $G$ such that $\alpha$ and $\beta$ does not commute . my first question is why $<\alpha,\beta>$ is of inifinite index in $G$ and my second question: Let $n$ be the least positive integer such that any subgroup generated by $n+1$ non commutative elements of $G$ is necessary of finite index in G what do we know about $n$ ?



non commutative elements in the fundamental group of a closed hyperbolic surface
The answer to the first question: because it is free, and every subgroup of finite index is the fundamental group of a finite cover of the surface (which is again a hyperbolic surface). The answer to the second question is "infinity". Take any free non-abelian subgroup $F$ of the surface group. The group $F$, being free, contains free subgroups $F\_n$ of arbitrary rank $n$. The generators of $F\_n$ do not commute and do not generate a subgroup of finite index. 


