Conjugator Length in Finitely Presented Groups

TL;DR

This work analyzes the conjugator length function in finitely presented groups, showing its spectrum contains all Dehn-function-type growth rates and, in particular, that for every $\alpha\ge 2$ computable in double-exponential time there exists a finitely presented group with $CL(n)\sim n^\alpha$. The authors develop and exploit $S$-machines to translate time functions into conjugator-length bounds, establishing a tight connection between time complexity of machine recognizers and conjugator-length growth via hubless group constructions and trapezium–computation correspondences. They prove a central result that $CL_{M(\mathbf{E}_{\mathbf{S}})}$ is asymptotically equivalent to the time function $TM_{\mathbf{S}}$, enabling the transfer of Dehn-function spectrum results to conjugator-length spectra and refining Bridson–Riley’s findings. The paper also analyzes the annular Dehn function, showing it can outpace the usual Dehn function, and demonstrates that there is no computable bound of the conjugator length in terms of the Dehn function, highlighting a deep separation between these invariants in finitely presented groups.

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 $α\geq2$ which is computable in double-exponential time, we show there exists a finitely presented group whose conjugator length function is asymptotically equivalent to $n^α$. 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.

Figures (5)

• Figure 4.1:
• Figure 4.2: $(\theta,q)$-annulus with defining $\theta$-band $\pazocal{T}$ and $q$-band $\pazocal{Q}$
• Figure 4.3: $\theta$-band $\pazocal{T}$ with trimmed top
• Figure 5.1:
• Figure 5.2: Spiral with $k=4$

Theorems & Definitions (105)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1: Lemma 2.1 of O18
• Lemma 2.2