Research in Geometry and Topology

尽管在自由群的自同构群的拓扑学上取得了重大进展，但对这个群的大规模几何却知之甚少。提案的一部分是一个程序，以弥补与映射类组的差异，其中几何是非常好的理解，感谢马苏尔，明斯基和其他人最近的工作。这个程序的第一步是自由因子复形的双曲性。其他问题包括Torelli群的有限可表示性、函数域上算术群的有限性以及直角Artin群的拟等距刚性。代数与几何之间的相互作用是数学中的经典主题之一。例如，柏拉图立体的对称性构成了群的基本例子。在几何群论中，情况正好相反：人们从一个代数对象的一组对称性（例如，另一组）开始，并构造一个具有相同对称性的几何对象。研究这个对象的几何形状，然后导致一个更好的理解代数对象之一开始。该提案载有这种做法的若干实例。Culler-Vogtmann的外层空间是一个几何对象，其对称性与自由群（有限秩）的对称性相同。最近的结果表明，这个空间在某些方向上具有负曲率，很像马鞍的表面。更好地理解这一现象将有助于更好地理解自由群的对称群。

In spite of a significant progress on the topology of the automorphism group of a free group, relatively little is known about the large scale geometry of this group. A part of the proposal is a program to remedy the discrepancy with mapping class groups, where the geometry is very well understood, thanks to the recent work of Masur, Minsky and others. The first step in this program is hyperbolicity of the free factor complex. Other questions proposed are finite presentability of Torelli groups, finiteness properties of arithmetic groups over function fields and quasi-isometric rigidity of right-angled Artin groups.The interplay between algebra and geometry is one of the classical themes in mathematics. For example, symmetries of Platonic solids form fundamental examples of groups. In geometric group theory the situation is reversed: one starts with a group of symmetries of an algebraic object (for example, another group) and constructs a geometric object with the same symmetries. The study of the geometry of this object then leads to a greater understanding of the algebraic object one started with. The proposal contains several instances of this approach. Culler-Vogtmann's Outer space is a geometric object whose symmetries are the same as those of a free group (of finite rank). Recent results reveal that this space has negative curvature in some directions, much like the surface of a saddle. Understanding this phenomenon better would lead to a greater understanding of the symmetry group of the free group.