(This question is reposted from math.SE, since it's sat there for a while with no answers. Apologies if it's not considered research-level, but I'm not a group theorist myself.)

Suppose I have a finite set of elements $x_1, \dots, x_n$ of the modular group $\operatorname{SL}_2(\mathbf{Z})$. Is there is an algorithm that will determine in finitely many steps whether or not the subgroup generated by $x_1, \dots, x_n$ has finite index?