Interactions between geometric group theory and topology

在几何群论中，人们将给定的群（代数对象）作为某个几何空间的对称性的集合来研究。这一观点已成功地应用于群从广泛的主题在数学，包括随机群，各种团体从低维拓扑，以及最近的克雷莫纳group.In这个项目中，我们感兴趣的群体是群同胚的表面。曲面是二维物体，包括球的表面、圆环面的表面和钻有孔的木块的表面。曲面的同胚是曲面的连续弯曲或拉伸，可以反转。同胚自然出现，例如，考虑混合下的流体表面。表面的同胚集合形成一个群。令人惊讶的是，很少有人知道这个群体，尽管它在数学中存在了100多年。一般来说，研究这些群体的工具很少。然而，Bowden、Hensel和Webb最近表明，只要表面上的“洞数”至少为1，这些群就可以用几何群论的方法来研究。使用的确切工具是构造一个（不可数无限）图，同胚群通过对称作用于该图，并且该图具有良好的几何形状（即格罗莫夫双曲几何）。这使得代数定理得以证明。不幸的是，这个新理论并没有以其最强的形式贯彻到二维球面的同胚群中。然而，如果我们把注意力限制在二维球面的保面积同胚上，那么类似的工具和构造可能会再次起作用。该项目的目标是排除这种结构的存在，或者证明并利用它来研究该群体。该项目属于EPSRC的“代数”和“几何与拓扑”研究领域。

In geometric group theory, one studies a given group (an algebraic object) as a collection of symmetries of some geometric space. This point of view has been applied successfully to groups from a wide range of topics in mathematics, including random groups, various groups from low-dimensional topology, and more recently the Cremona group.In this project, the groups that we are interested in are the groups of homeomorphisms of surfaces. Surfaces are 2-dimensional objects, and include the surface of a ball, the surface of a ring doughnut, and the surface of a block of wood with holes drilled through it. A homeomorphism of a surface is a continuous bending or stretching of the surface, which can be reversed. Homeomorphisms appear naturally, for instance, consider the surface of a fluid under mixing.The collection of homeomorphisms of a surface forms a group. Surprisingly very little is known about this group, despite its presence in mathematics for more than 100 years. There are in general few tools to study these groups. However Bowden, Hensel, and Webb recently showed that, provided that the "number of holes" in the surface is at least 1, these groups can be studied using techniques from geometric group theory. The exact tool used was the construction of an (uncountably infinite) graph on which the homeomorphism group acts by symmetries, and such that the graph has nice geometry (namely Gromov hyperbolic geometry). This enables algebraic theorems to be proved. Unfortunately this new theory does not carry through in its strongest form for the homeomorphism group of the 2-dimensional sphere. However if we restrict attention to the area-preserving homeomorphisms of the 2-dimensional sphere, then it might be the case that similar tools and constructions will work once again. The goal of the project is to either rule out the existence of such a construction, or prove and utilize it to study the group.This project lies in the EPSRC research areas of "Algebra" and "Geometry and Topology".