Twisted conjugacy classes in Lie groups
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910114134360064 |
|---|---|
| author | Prakash, Ravi Shah, Riddhi |
| author_facet | Prakash, Ravi Shah, Riddhi |
| contents | We consider twisted conjugacy classes of continuous automorphisms $φ$ of a Lie group $G$. We obtain a necessary and sufficient condition on $φ$ for its Reidemeister number, the number of twisted conjugacy classes, to be infinite when $G$ is connected and solvable or compactly generated and nilpotent. We also show for a general connected Lie group $G$ that the number of conjugacy classes is infinite. We prove that for a connected non-nilpotent Lie group $G$, there exists $n\in \mathbb{N}$ such that Reidemeister number of $φ^n$ is infinite for every $φ$. We say that $G$ has topological $R_\infty$-property if the Reidemeister number of every $φ$ is infinite. We obtain conditions on a connected solvable Lie group under which it has topological $R_\infty$-property; which, in particular, enables us to prove that the group of invertible $n\times n$ upper triangular real matrices and its quotient group modulo its center have topological $R_\infty$-property for every $n\geq 2$. We also prove that the Walnut group also has this property. We show that ${\mathrm{SL}}(2,\mathbb{R})$ and ${\mathrm{GL}}(2,\mathbb{R})$ have topological $R_\infty$-property, and construct many examples of Lie groups with this property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17927 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twisted conjugacy classes in Lie groups Prakash, Ravi Shah, Riddhi Group Theory Primary: 22E15, 22D45. Secondary: 22E25 We consider twisted conjugacy classes of continuous automorphisms $φ$ of a Lie group $G$. We obtain a necessary and sufficient condition on $φ$ for its Reidemeister number, the number of twisted conjugacy classes, to be infinite when $G$ is connected and solvable or compactly generated and nilpotent. We also show for a general connected Lie group $G$ that the number of conjugacy classes is infinite. We prove that for a connected non-nilpotent Lie group $G$, there exists $n\in \mathbb{N}$ such that Reidemeister number of $φ^n$ is infinite for every $φ$. We say that $G$ has topological $R_\infty$-property if the Reidemeister number of every $φ$ is infinite. We obtain conditions on a connected solvable Lie group under which it has topological $R_\infty$-property; which, in particular, enables us to prove that the group of invertible $n\times n$ upper triangular real matrices and its quotient group modulo its center have topological $R_\infty$-property for every $n\geq 2$. We also prove that the Walnut group also has this property. We show that ${\mathrm{SL}}(2,\mathbb{R})$ and ${\mathrm{GL}}(2,\mathbb{R})$ have topological $R_\infty$-property, and construct many examples of Lie groups with this property. |
| title | Twisted conjugacy classes in Lie groups |
| topic | Group Theory Primary: 22E15, 22D45. Secondary: 22E25 |
| url | https://arxiv.org/abs/2508.17927 |