The lengths of conjugators in the model filiform groups
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914316861571072 |
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| author | Bridson, Martin R. Riley, Timothy R. |
| author_facet | Bridson, Martin R. Riley, Timothy R. |
| contents | The conjugator length function of a finitely generated group $Γ$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $Γ$. We prove that polynomials of arbitrary degree arise as conjugator length functions of finitely presented groups. To establish this, we analyse the geometry of conjugation in the discrete model filiform groups $Γ_d = \mathbb{Z}^d\rtimes_ϕ\mathbb{Z}$ where is $ϕ$ is the automorphism of $\mathbb{Z}^d$ that fixes the last element of a basis $a_1,\dots,a_d$ and sends $a_i$ to $a_ia_{i+1}$ for $i<d$. The conjugator length function of $Γ_d$ is polynomial of degree $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01235 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The lengths of conjugators in the model filiform groups Bridson, Martin R. Riley, Timothy R. Group Theory 20F65, 20F10, 20F18 The conjugator length function of a finitely generated group $Γ$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $Γ$. We prove that polynomials of arbitrary degree arise as conjugator length functions of finitely presented groups. To establish this, we analyse the geometry of conjugation in the discrete model filiform groups $Γ_d = \mathbb{Z}^d\rtimes_ϕ\mathbb{Z}$ where is $ϕ$ is the automorphism of $\mathbb{Z}^d$ that fixes the last element of a basis $a_1,\dots,a_d$ and sends $a_i$ to $a_ia_{i+1}$ for $i<d$. The conjugator length function of $Γ_d$ is polynomial of degree $d$. |
| title | The lengths of conjugators in the model filiform groups |
| topic | Group Theory 20F65, 20F10, 20F18 |
| url | https://arxiv.org/abs/2506.01235 |