For a locally compact group G, we study the distality of the action of automorphisms T of G on SubG, 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 SubG if and only if Tn 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 SubG if and only if it acts distally on SubΓ. This also holds for any closed T-invariant co-compact subgroup Γ in G. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on SubΓ. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group and that in a nilpotent Lie group. We also characterise automorphisms of a lattice Γ in a connected semisimple Lie group which act distally on SubΓ. For torsion-free compactly generated nilpotent (metrisable) groups G, we obtain the following characterisation: T acts distally on SubG if and only if T is contained in a compact subgroup of Aut(G). Using these results, we characterise the class of such groups G which act distally on SubG. We also show that any compactly generated distal group G is Lie projective.

中文翻译：

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幂等群 G 上的自同态的远端作用及其在李群中的格的应用

对于局部紧群G，我们研究了自同构作用的远端性吨的G在子上G, 的闭子群的紧空间G具有 Chabauty 拓扑。对于某一类离散组G, 我们证明吨远端作用于 SubG当且仅当吨n是某些人的身份图$n\in\mathbb N$. 作为一个应用程序，我们得到一个吨- 单连通幂零李群中的不变格 ΓG,吨远端作用于 SubG当且仅当它远端作用于 SubΓ. 这也适用于任何关闭吨-不变的协紧子群 Γ inG. 对于简单连通的可解李群中的格 Γ，我们研究了它的自同构远端作用于 Sub 的条件Γ. 我们构建了一个示例，突出显示可解李群中格上的自同构行为与幂零李群中的行为之间的差异。我们还描述了一个连接的半单李群中格子Γ的自同构，它在远端上作用于 SubΓ. 对于无扭紧生成的幂零（可度量）群G，我们得到以下特征：吨远端作用于 SubG当且仅当吨包含在 Aut(G）。使用这些结果，我们描述了这些组的类别G远端作用于 SubG. 我们还表明，任何紧凑生成的远端组G是李射影。