arXiv math/96072211996-07-01math.RT

可解李群的超刚性子群

Superrigid subgroups of solvable Lie groups

Dave Witte

摘要：

设$Γ$是单连通可解李群~ $G$的离散子群，使得$\Ad_GΓ$与$\Ad G$具有相同的Zerkki闭包.如果$α\colon Γ\to \GL_n(\real)$是~ $Γ$的任意有限维表示，我们证明了$α$实际上扩展到~ $G$的连续表示~ $σ$.进一步地，~ $σ$的象包含在~ $α$的象的Zapriki闭包中.当$Γ$不是离散的时，如果我们另外假设$[Γ, Γ]$的闭包是$[G,G] \cap Γ$的有限指数子群（并且$Γ$是闭的，而$α$是连续的），同样的结论也成立。

英文摘要：

Let $Γ$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_GΓ$ has the same Zariski closure as $\Ad G$. If $α\colon Γ\to \GL_n(\real)$ is any finite-dimensional representation of~$Γ$,we show that $α$ virtually extends to a continuous representation~$σ$ of~$G$. Furthermore, the image of~$σ$ is contained in the Zariski closure of the image of~$α$. When $Γ$ is not discrete, the same conclusions are true if we make the additional assumption that the closure of $[Γ, Γ]$ is a finite-index subgroup of $[G,G] \cap Γ$ (and $Γ$ is closed and $α$ is continuous).

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