Free subgroups of free products and combinatorial hypermaps

TL;DR

This paper establishes a connection between free subgroups of a free product of cyclic groups and hypermaps, deriving generating series, recurrence relations, and asymptotic formulas for counting these subgroups and hypermaps.

Contribution

It introduces a novel approach linking free subgroup enumeration to hypermap theory, providing explicit generating functions and asymptotic analysis.

Findings

Generated a transcendental series for subgroup counts.

Derived nonlinear recurrence relations from Riccati-type differential equations.

Provided asymptotic formulas for subgroup and hypermap enumeration.

Abstract

We derive a generating series for the number of free subgroups of finite index in $\Delta^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $\Delta^+$ 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 $\Delta^+$, 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.