Conjugator Length in Finitely Presented Groups

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.

💡 Research Summary

The paper investigates the range of functions that can appear as the conjugator length function (CL) of finitely presented groups. The conjugator length function measures, for a given bound n, the maximal minimal length of a word γ that conjugates any pair of group elements represented by words of total length at most n. This function captures the intrinsic difficulty of the conjugacy problem in a geometric way, analogous to how the Dehn function captures the difficulty of the word problem.

The authors begin by recalling the classical Dehn, conjugacy, and isomorphism problems introduced by Max Dehn, and they explain how the Dehn function provides an isoperimetric invariant for finitely presented groups. Earlier work by Bridson and Riley showed that for every integer m there exists a finitely presented group G with CL_G(n) ≍ n^m, and later they proved that the set of exponents α for which n^α is equivalent to a conjugator length function is dense in the rational interval