Superexponential distortion of free groups in virtually special Groups

TL;DR

This paper constructs examples of virtually special groups with free subgroups exhibiting extremely rapid distortion growth, including superexponential and beyond, demonstrating complex subgroup embedding behaviors.

Contribution

It introduces new constructions of virtually special groups with free subgroups having superexponential and faster distortion functions, expanding understanding of subgroup distortion complexity.

Findings

Constructed groups with distortion functions like xp^k(x^m)

Provided examples with distortion exceeding any iterated exponential

Demonstrated the existence of highly distorted free subgroups in virtually special groups

Abstract

For all integers $k, m > 0$, we construct a virtually special group $G$ containing a finite rank free subgroup $F$ whose distortion function in $G$ grows like $\exp^k(x^m)$. We also construct examples of virtually special groups containing finite rank free subgroups whose distortion functions grow bigger than any iterated exponential.