Membership Problem for the Modular Group

The modular group plays an important role in many branches of mathematics. We show that the membership problem for the modular group is decidable in polynomial time. To this end, we develop a new syllable-based version of the known subgroup-graph approach. The new approach can be used to prove additional results. We demonstrate this by using it to prove that the membership problem for a free group remains decidable in polynomial time when elements are written in a normal form with exponents.