Interactions between geometric group theory and topology

在几何群论中，人们将一个给定的群(一个代数对象)研究为某个几何空间的对称性的集合。这一观点已经成功地应用于数学中的各种群，包括随机群、低维拓扑的各种群，以及最近的Cremona群。在这个项目中，我们感兴趣的群是曲面的同胚群。曲面是二维对象，包括球的表面、环形甜甜圈的表面和带有孔的木块的表面。曲面的同胚是曲面的连续弯曲或拉伸，可以反转。同态是自然出现的，例如，考虑流体在混合中的表面。一个表面的同态集合形成一个群。令人惊讶的是，人们对这个群体知之甚少，尽管它在数学中已经存在了100多年。一般来说，研究这些群体的工具很少。然而，鲍登、亨塞尔和韦伯最近证明，只要表面上的“洞的数量”至少为1，就可以用几何群论中的技术来研究这些群。所使用的确切工具是构造一个(不可数无限的)图，在该图上同胚群通过对称性作用，并且使得该图具有良好的几何(即Gromov双曲几何)。这使得代数定理能够被证明。不幸的是，这一新理论并没有以最强的形式适用于2维球体的同胚群。然而，如果我们将注意力局限于2维球体的保面积同胚，那么类似的工具和构造可能会再次起作用。该项目的目标是要么排除这种结构的存在，要么证明并利用它来研究这个群体。该项目位于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".