Short Curves on Flat Surfaces and the Curve Graph

Dr. Robert Tang Sydney University


On a flat torus, any simple closed curve has an arbitrarily short geodesic representative upon applying a suitable SL(2,R)-deformation to the flat metric. For a given flat metric on higher genus surface, however, the same is not true. We will study the set of curves which can be made short on some metric obtainable under some SL(2,R)-deformation. This "short curve" set has some interesting coarse geometric properties when viewed as a subset of the curve graph - a combinatorial tool which has been used to study mapping class groups, Teichmuller space and hyperbolic 3-manifolds. I will also describe nearest point projection properties to the short curve set within the curve graph. This is a joint work with Richard Webb.

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