Geometric Methods in Group Theory

摘要，巴斯，9700634该项目将涉及几何群论的三个领域。第一类是关于树上的群作用，特别是树格，即具有有限体积商的局部有限树的自同构离散群。这个理论是作用在上半平面上的富氏群理论的离散类比。第二个领域是有限生成群的表示理论，特别是刚性群的结构，即那些在每个维度上只有有限多个不可约表示的群。如果也是线性的，则推测它们是算术类型的。第三个领域涉及仿射空间上的代数群作用，特别是约化群的平坦族表示，以及这些表示在多大程度上是全局等变向量丛。这个项目是对各种几何结构的某些对称性群的研究。在第一个例子中，这些是无限树(即没有闭合回路的连通图)，我们的对称群是离散的，并以有限多个轨道起作用。在第二种情况下，我们通过有限生成群的矩阵表示来研究有限生成群，特别是对于那些“很少”生成有限生成群的群。在最后一种情况下，我们研究李群在欧几里德空间上的代数作用，但可能是非线性作用，并试图确定某些这样的作用何时是“近似线性的”。

Abstract, Bass, 9700634 The project will address three areas of geometric group theory. The first concerns group actions on trees, particularly tree lattices, i.e. discrete groups of automorphisms of locally finite trees with finite volume quotients. This theory is a discrete analogue of the theory of Fuchsian groups acting on the upper half plane. The second area is the representation theory of finitely generated groups, and particularly the structure of rigid groups, i.e. those with only finitely many irreducible representations in each dimension. When also linear, these are conjecturally of arithmetic type. The third area concerns algebraic group actions on affine space, and, in particular, flat families of representations of reductive groups, and the extent to which these are globally equivariant vector bundles. The project is an investigation of certain groups of symmetries of various kinds of geometric structures. In the first instance these are infinite trees, ( i.e. connected graphs without closed circuits), and our symmetry groups are discrete, and act with say finitely many orbits. In the second setting, we study finitely generated groups via their matrix representations, particularly for groups that have "very few" of these. In the last case we study Lie groups acting algebraically, but possibly non linearly, on Euclidean spaces, and try to determine when certain such actions are "approximately linear."