Critical exponents of boomerang subgroups in the free group

TL;DR

This paper constructs specific subgroups within free groups acting on Cayley trees, demonstrating that their critical exponents can be made arbitrarily close to those of any finitely generated subgroup, revealing new insights into subgroup dynamics.

Contribution

It introduces a method to construct boomerang subgroups with critical exponents approaching those of arbitrary finitely generated subgroups in free groups.

Findings

Existence of boomerang subgroups with arbitrary close critical exponents

New techniques for subgroup construction in free groups

Insights into the relationship between subgroup structure and critical exponents

Abstract

We construct, for the free group acting on its Cayley tree, boomerang subgroups whose critical exponent is arbitrarily close to the critical exponent of a given finitely generated subgroup.