Growth and cogrowth of normal subgroups of a free group

TL;DR

This paper establishes a condition based on the planarity of quotient graphs that ensures a sequence of normal subgroups in a free group has both growth and cogrowth approaching their extremal bounds.

Contribution

It introduces a planarity-based criterion that guarantees simultaneous extremal growth and cogrowth behaviors in sequences of normal subgroups of free groups.

Findings

Growths tend to the upper bound

Cogrowths tend to the lower bound

Planarity of quotient graphs is the key condition

Abstract

We give a sufficient condition for a sequence of normal subgroups of a free group to have the property that both, their growths tend to the upper bound and their cogrowths tend to the lower bound. The condition is represented by planarity of the quotient graphs of the tree.