Groups with fast-growing conjugator length functions

TL;DR

This paper constructs finitely presented groups with conjugator length functions that grow exponentially and even faster, reaching the levels of the Grzegorczyk hierarchy, revealing new growth behaviors in group theory.

Contribution

It provides the first examples of finitely presented groups with conjugator length functions exhibiting exponential and super-exponential growth rates.

Findings

Constructed groups with exponential conjugator length functions.

Developed a family of groups with conjugator length growth matching Grzegorczyk hierarchy levels.

Demonstrated the diversity of growth behaviors in finitely presented groups.

Abstract

We construct the first examples of finitely presented groups where the conjugator length function is exponential; these are central extensions of groups of the form $F_m \rtimes F_2$. Further, we use a fibre product construction to exhibit a family of finitely presented groups $\Gamma_k$ where, for each $k$, the conjugator length function of $\Gamma_k$ grows like functions in the $k$-th level of the Grzegorczyk hierarchy of primitive recursive functions.