Geometric Methods in Group Theory

摘要，低音，9700634 该项目将解决几何群论的三个领域。 第一个涉及树上的群作用，特别是树格，即具有有限体积子的局部有限树的自同构的离散群。 这个理论是一个离散的模拟理论的Fuchsian集团的上半平面。 第二个领域是群生成的表示理论，特别是刚性群的结构，即那些在每个维度上只有1000个不可约表示的群。 当也是线性的时，它们在理论上是算术类型的。 第三个领域涉及代数群行动仿射空间，特别是，平面家庭的代表性的还原组，以及在何种程度上，这些是全球等变向量束。 该项目是对各种几何结构的某些对称群的研究。 在第一种情况下，这些是无限树（即没有闭合回路的连通图），并且我们的对称群是离散的，并且具有例如1000个轨道。 在第二种情况下，我们研究了通过矩阵表示的群，特别是对于那些“很少”的群。 在最后一种情况下，我们研究李群在欧几里得空间上的代数作用，但可能是非线性的，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."