we will survey the class of linear groups, and identify three methods of constructing linear groups: algebraic, geometric, and arithmetic. we will propose that every subgroup comes from one of these types of constructions. there are many interesting consequences all across group theory and geometry if this is indeed the case. focusing on the case of 2-by-2 matrix groups, the algebraic and arithmetic subgroups are very well-understood. we will fill in some understanding of geometric subgroups in this setting, by showing that for any prime p, there is a closed surface group in \({\rm psl}_2(\mathbb{z}[1/p])\). these surface groups will have genus proportional to \(p\), and we speculate that the construction is optimal.