Conjugator Length in Finitely Presented Groups

TL;DR

This paper explores the possible growth rates of conjugator length functions in finitely presented groups, showing they can match any computable Dehn function, including functions like $n^eta$, and investigates their relationships with other group-theoretic functions.

Contribution

It demonstrates that conjugator length functions can realize any sufficiently large computable function, refining previous results and establishing a connection with Dehn functions via $S$-machines.

Findings

Conjugator length functions can match any computable Dehn function.

Existence of finitely presented groups with conjugator length asymptotic to $n^eta$ for computable $eta$.

The relationship between conjugator length, Dehn, and annular Dehn functions is clarified.

Abstract

The conjugator length function of a finitely generated group is the function $f$ so that $f(n)$ is the minimal upper bound on the length of a word realizing the conjugacy of two words of length at most $n$. We study herein the spectrum of functions which can be realized as the conjugator length function of a finitely presented group, showing that it contains every function that can be realized as the Dehn function of a finitely presented group. In particular, given a real number $\alpha\geq2$ which is computable in double-exponential time, we show there exists a finitely presented group whose conjugator length function is asymptotically equivalent to $n^\alpha$. This yields a substantial refinement to results of Bridson and Riley. We attain this result through the computational model of $S$-machines, achieving the more general result that any sufficiently large function which can be realized as the time function of an $S$-machine can also be realized as the conjugator length function of a finitely presented group. Finally, we use the constructed groups to explore the relationship between the conjugator length function, the Dehn function, and the annular Dehn function in finitely presented groups.