Summary
Given a mathematical object, a common theme is to study the symmetries of that object. In this project, the objects are compact topological surfaces, and the group of symmetries is the mapping class group.
In this project, we will investigate simplicial graphs associated to surfaces, which have proved to be key tools in the study of both the algebraic and the geometric structure of mapping class groups. Studying the geometry of groups has proved to be a profound way to study their algebraic properties. We will focus on a graph called the pants graph, whose vertices represent pants decompositions of the surface (collections of homotopy classes of simple closed curves that cut the surface into spheres with three holes). The pants graph is significant not only in the study of mapping class groups, but also in studying the hyperbolic geometry of surfaces and 3-manifolds.
The first part of the project is to understand how distances between vertices in the pants graph are related to the number of intersections between the corresponding pants decompositions. For a related graph, the curve graph, it is known that the distance between two vertices is bounded above by a logarithmic function of the number of intersections, but the methods do not immediately generalise to the pants graph. We will also investigate questions of computational complexity around computing distances in the pants graph. This part of the project will include a secondment at a computer science department.
The second part of the project is to investigate maps from the pants graph to itself which preserve distances up to bounded error (such maps are called quasi-isometries). In a general metric space, the group of quasi-isometries is much bigger than the isometry group, but for most pants graphs, Bowditch proved that the two groups coincide, a property called quasi-isometric rigidity. We aim to prove that the same is true for three of the remaining unsolved cases.
In this project, we will investigate simplicial graphs associated to surfaces, which have proved to be key tools in the study of both the algebraic and the geometric structure of mapping class groups. Studying the geometry of groups has proved to be a profound way to study their algebraic properties. We will focus on a graph called the pants graph, whose vertices represent pants decompositions of the surface (collections of homotopy classes of simple closed curves that cut the surface into spheres with three holes). The pants graph is significant not only in the study of mapping class groups, but also in studying the hyperbolic geometry of surfaces and 3-manifolds.
The first part of the project is to understand how distances between vertices in the pants graph are related to the number of intersections between the corresponding pants decompositions. For a related graph, the curve graph, it is known that the distance between two vertices is bounded above by a logarithmic function of the number of intersections, but the methods do not immediately generalise to the pants graph. We will also investigate questions of computational complexity around computing distances in the pants graph. This part of the project will include a secondment at a computer science department.
The second part of the project is to investigate maps from the pants graph to itself which preserve distances up to bounded error (such maps are called quasi-isometries). In a general metric space, the group of quasi-isometries is much bigger than the isometry group, but for most pants graphs, Bowditch proved that the two groups coincide, a property called quasi-isometric rigidity. We aim to prove that the same is true for three of the remaining unsolved cases.
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| Web resources: | https://cordis.europa.eu/project/id/101106914 |
| Start date: | 01-09-2023 |
| End date: | 31-08-2025 |
| Total budget - Public funding: | - 175 920,00 Euro |
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Original description
Given a mathematical object, a common theme is to study the symmetries of that object. In this project, the objects are compact topological surfaces, and the group of symmetries is the mapping class group.In this project, we will investigate simplicial graphs associated to surfaces, which have proved to be key tools in the study of both the algebraic and the geometric structure of mapping class groups. Studying the geometry of groups has proved to be a profound way to study their algebraic properties. We will focus on a graph called the pants graph, whose vertices represent pants decompositions of the surface (collections of homotopy classes of simple closed curves that cut the surface into spheres with three holes). The pants graph is significant not only in the study of mapping class groups, but also in studying the hyperbolic geometry of surfaces and 3-manifolds.
The first part of the project is to understand how distances between vertices in the pants graph are related to the number of intersections between the corresponding pants decompositions. For a related graph, the curve graph, it is known that the distance between two vertices is bounded above by a logarithmic function of the number of intersections, but the methods do not immediately generalise to the pants graph. We will also investigate questions of computational complexity around computing distances in the pants graph. This part of the project will include a secondment at a computer science department.
The second part of the project is to investigate maps from the pants graph to itself which preserve distances up to bounded error (such maps are called quasi-isometries). In a general metric space, the group of quasi-isometries is much bigger than the isometry group, but for most pants graphs, Bowditch proved that the two groups coincide, a property called quasi-isometric rigidity. We aim to prove that the same is true for three of the remaining unsolved cases.
Status
SIGNEDCall topic
HORIZON-MSCA-2022-PF-01-01Update Date
31-07-2023
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