Interactions between geometric group theory and topology
Lead Research Organisation:
University of Manchester
Department Name: Mathematics
Abstract
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".
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".
Organisations
People |
ORCID iD |
| Richard Webb (Primary Supervisor) | |
| Yongsheng Jia (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/W523884/1 | 01/10/2021 | 30/09/2025 | |||
| 2625336 | Studentship | EP/W523884/1 | 01/10/2021 | 31/03/2025 | Yongsheng Jia |