Free subgroups of free products and combinatorial hypermaps

We derive a generating series for the number of free subgroups of finite index in ${\mathrm{\Delta }}^{+}={\mathbb{Z}}_p\ast {\mathbb{Z}}_q$ by using a connection between free subgroups of ${\mathrm{\Delta }}^{+}$ and certain hypermaps (also known as ribbon graphs or “fat” graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy. We also study the generating series for conjugacy classes of free subgroups of finite index in ${\mathrm{\Delta }}^{+}$, which correspond to isomorphism classes of hypermaps. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, or, equivalently, various types of hypermaps and their isomorphism classes.