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Department of Mathematics pres.

Topology Seminar

Extremal length calculations and applications to the geodesic flow

A collar neighborhood of a simple closed geodesic in a hyperbolic surface is an open neighborhood of the geodesic which is topologically an annulus. It is well-known that a simple closed geodesic on a hyperbolic surface has a natural (or standard) collar. The outstanding feature of the natural collar is that its size depends on local data, namely its size depends only on the length of the geodesic. Using this collar one can make extremal length calculations of curve families that are transverse to the geodesic.

In this talk, after defining extremal length and discussing its properties, we define a new type of collar which we call a non-standard collar. Using the non-standard collar we are able to improve estimates on the extremal length of curve families that are transverse to the geodesic and give a number of applications to the geodesic flow on an infinite type hyperbolic surface. This is joint work with Hrant Hakobyan and Dragomir Saric. Speaker(s): Ara Basmajian (CUNY)

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