Conjugator Length in Finitely Presented Groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912886623830016 |
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| author | Gillis, Conan Wagner, Francis |
| author_facet | Gillis, Conan Wagner, Francis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08053 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Conjugator Length in Finitely Presented Groups Gillis, Conan Wagner, Francis Group Theory 20F65, 20F10, 20F05, 20F06, 03D10, 03D15, 20F69 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. |
| title | Conjugator Length in Finitely Presented Groups |
| topic | Group Theory 20F65, 20F10, 20F05, 20F06, 03D10, 03D15, 20F69 |
| url | https://arxiv.org/abs/2601.08053 |