Firstly, we derive a generating series for the number of free subgroups offinite index in $Delta^+ = mathbb{Z}_2*mathbb{Z}_2*mathbb{Z}_2$ by using aconnection between free subgroups of $Delta^+$ and certain three dimensionalmaps 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 freesubgroups of finite index in $Delta^+$, which correspond to isomorphismclasses of pavings. Asymptotic formulas are provided for the numbers of freesubgroups of given finite index, conjugacy classes of such subgroups, and theequivalent types of pavings and their isomorphism classes.
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