论文标题
SUB $ _g $的离散组自动形态行动的动力和lie groups的应用程序的应用
Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub$_G$ and Applications to Lattices in Lie Groups
论文作者
论文摘要
对于离散的集团$ g $和紧凑型空间$ _g $ _g $（封闭式）的$ g $的亚组，赋予了Chabauty拓扑结构，我们研究了在远距离和扩张性方面，$ g $ $ _g $ $ g $的行动动态。我们还研究了连接的谎言组中晶格$γ$的结构和特性。特别是，我们表明，$γ$的唯一最大可溶解的正常亚组是多环状，$γ$的相应商是有限的，要么是乘坐cofinite子群，该子组是具有某些特性的连接的半圣经谎言组中的晶格。我们还表明，$γ$的循环子组集sub $^c_γ$在$_γ$中关闭。我们证明，无限的离散组$γ$，它是连接的谎言组中的多环子或晶格，不承认任何自动形态广泛作用于sub $^c_γ$，而只有$γ$的有限订单自动形态自动形态在sub $^c_γ$上行动。对于连接的Lie Group $ g $的自动形态$ T $和$ g $中的$ t $ invariant lattice $γ$，我们比较了Sub $ _g $和SUB $ _G $和SUB $_γ$的行为。我们将某些条件放在Lie Group $ G $的结构上，根据该结构，我们表明$ T $在Sub $ _g $时远端起作用，并且仅当它在sub $_γ$上以远端作用。我们构建反示例，以表明如果谎言组的条件放松，这通常不存在。
For a discrete group $G$ and the compact space Sub$_G$ of (closed) subgroups of $G$ endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of $G$ on Sub$_G$ in terms of distality and expansivity. We also study the structure and properties of lattices $Γ$ in a connected Lie group. In particular, we show that the unique maximal solvable normal subgroup of $Γ$ is polycyclic and the corresponding quotient of $Γ$ is either finite or admits a cofinite subgroup which is a lattice in a connected semisimple Lie group with certain properties. We also show that Sub$^c_Γ$, the set of cyclic subgroups of $Γ$, is closed in Sub$_Γ$. We prove that an infinite discrete group $Γ$ which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on Sub$^c_Γ$, while only the finite order automorphisms of $Γ$ act distally on Sub$^c_Γ$. For an automorphism $T$ of a connected Lie group $G$ and a $T$-invariant lattice $Γ$ in $G$, we compare the behaviour of the actions of $T$ on Sub$_G$ and Sub$_Γ$ in terms of distality. We put certain conditions on the structure of the Lie group $G$ under which we show that $T$ acts distally on Sub$_G$ if and only if it acts distally on Sub$_Γ$. We construct counter examples to show that this does not hold in general if the conditions on the Lie group are relaxed.
