Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866914822161956864 |
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| author | Palit, Rajdip Shah, Riddhi |
| author_facet | Palit, Rajdip Shah, Riddhi |
| contents | For a locally compact group $G$, we study the distality of the action of automorphisms $T$ of $G$ on ${\rm Sub}_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. For a certain class of discrete groups $G$, we show that $T$ acts distally on ${\rm Sub}_G$ if and only if $T^n$ is the identity map for some $n\in{\mathbb N}$. As an application, we get that for a $T$-invariant lattice $Γ$ in a simply connected nilpotent Lie group $G$, $T$ acts distally on ${\rm Sub}_G$ if and only if it acts distally on ${\rm Sub}_Γ$. This also holds for any closed $T$-invariant co-compact subgroup $Γ$. For a lattice $Γ$ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on ${\rm Sub}_Γ$. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group from that in a nilpotent Lie group. For torsion-free compactly generated nilpotent (metrizable) groups $G$, we obtain the following characterisation: $T$ acts distally on ${\rm Sub}_G$ if and only if $T$ is contained in a compact subgroup of ${\rm Aut}(G)$. Using these results, we characterise the class of such groups $G$ which act distally on ${\rm Sub}_G$. We also show that any compactly generated distal group $G$ is Lie projective. As a consequence, we get some results on the structure of compactly generated nilpotent groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_01446 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups Palit, Rajdip Shah, Riddhi Dynamical Systems Group Theory For a locally compact group $G$, we study the distality of the action of automorphisms $T$ of $G$ on ${\rm Sub}_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. For a certain class of discrete groups $G$, we show that $T$ acts distally on ${\rm Sub}_G$ if and only if $T^n$ is the identity map for some $n\in{\mathbb N}$. As an application, we get that for a $T$-invariant lattice $Γ$ in a simply connected nilpotent Lie group $G$, $T$ acts distally on ${\rm Sub}_G$ if and only if it acts distally on ${\rm Sub}_Γ$. This also holds for any closed $T$-invariant co-compact subgroup $Γ$. For a lattice $Γ$ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on ${\rm Sub}_Γ$. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group from that in a nilpotent Lie group. For torsion-free compactly generated nilpotent (metrizable) groups $G$, we obtain the following characterisation: $T$ acts distally on ${\rm Sub}_G$ if and only if $T$ is contained in a compact subgroup of ${\rm Aut}(G)$. Using these results, we characterise the class of such groups $G$ which act distally on ${\rm Sub}_G$. We also show that any compactly generated distal group $G$ is Lie projective. As a consequence, we get some results on the structure of compactly generated nilpotent groups. |
| title | Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups |
| topic | Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/1912.01446 |