Three-dimensional maps and subgroup growth

Laura Ciobanu, Alexander Kolpakov

Published 2017-12-04Version 1

Firstly, we derive a generating series for the number of free subgroups of finite index in $\Delta^+ = \mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z}_2$ by using a connection between free subgroups of $\Delta^+$ and certain three dimensional maps known as pavings, and show that this generating series is non-holonomic. We also provide a non-linear recurrence relation for its coefficients. Secondly, we study the generating series for conjugacy classes of free subgroups of finite index in $\Delta^+$, which correspond to isomorphism classes of pavings. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, and the equivalent types of pavings and their isomorphism classes.