2010 Fiscal Year Final Research Report
Combinatorics in Topology
| Project/Area Number |
21740040
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
Geometry
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| Research Institution | The University of Tokyo |
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Principal Investigator |
SHACKLETON Kenneth The University of Tokyo, 数物連携宇宙研究機構, 特任研究員 (70536870)
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| Project Period (FY) |
2009 – 2010
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| Keywords | 位相幾何 |
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| Research Abstract |
Our research has focused primarily on the geometry of the pants graph. The vertices and edges of the pants graph derive from elementary objects and intersection relations on a surface. When we assign length one to each edge, we have a metric space. The most startling property of this space is a theorem of Brock's that it "coarsely" models the Weil-Petersson (WP) metric on Teichmueller space. That is, in sacrificing much of the original structure of the WP-metric we can naturally arrive at the pants graph. The overriding question then becomes: To what extent does the pants graph model the WP-metric- For example, which geometric properties of WP have full geometric analogues in the pants graph?
This is widely considered a pertinent and challenging problem for which we have been able to make the most meaningful progress: When the surface is the 5-holed sphere, we have shown that any two distinct ideal points are connected by a bi-infinite geodesic, that every pseudo-Anosov mapping class (when raised to a sufficiently high power) has a geodesic axis, and that hierarchy path systems in the curve graph often describe shortest paths in the pants graph. These are WP-geometric analogues. This work has just been accepted for publication after peer-review with no corrections necessary.
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Research Products
(7 results)
