Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911143284441088 |
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| author | Chatterjee, Debamita Shah, Riddhi |
| author_facet | Chatterjee, Debamita Shah, Riddhi |
| contents | For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the action of $T$ on Sub$_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. We study the action of $T$ on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of $G$. We relate the distality of the $T$-action on Sub$^p_G$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of the closed subgroup generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central torus and $G$ is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$, and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group $G$ acts distally on Sub$^p_G$ by conjugation if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. Some of our results extend those of Shah and Yadav. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups Chatterjee, Debamita Shah, Riddhi Dynamical Systems Group Theory For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the action of $T$ on Sub$_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. We study the action of $T$ on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of $G$. We relate the distality of the $T$-action on Sub$^p_G$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of the closed subgroup generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central torus and $G$ is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$, and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group $G$ acts distally on Sub$^p_G$ by conjugation if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. Some of our results extend those of Shah and Yadav. |
| title | Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups |
| topic | Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2406.01237 |