The Proalgebraic Completion of Rigid Groups
The Proalgebraic Completion of Rigid Groups
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刚性群的前代数完备性
DOI：
10.1023/a:1021221727311
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发表时间：
2002
影响因子：
0.5
通讯作者：
S. Mozes
中科院分区：
文献类型：
--
作者：
H. Bass;A. Lubotzky;A. Magid;S. Mozes
如果对于每个 n，有限生成的群 Γ 被称为刚性表示（简称为刚性表示），则 Γ 在 n 维上仅具有有限多类简单 ℂ 表示。示例包括更高等级的 S 算术组。通过Margulis超刚度，后者具有更强的性质：它们是表征超刚度；即，它们的前代数完备性是有限维的。我们构建了非线性刚性群的示例，这些示例不是超刚性的，并且表现出无限维数的每种可能类型。线性表示刚性群是否超刚性仍然是一个悬而未决的问题。
A finitely generated group Γ is called representation rigid (briefly, rigid) if for every n, Γ has only finitely many classes of simple ℂ representations in dimension n. Examples include higher rank S-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.