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2020-02-02 10:15:51 (UTC)

Incompressible

A subsurface S' of a surface S is incompressible if each essential loop in S' is still essential in S. Note that, an essential loop is a loop which is not homotopic to a point. I will try to give some examples and counterexamples without drawing anything since I could not find a way to attach pictures on this website. On a hyperbolic surface, we suppose that there is a hyperbolic one-holed torus embedded on the surface. Then the one-holed torus is incompressible. A hyperbolic pair of pants on a hyperbolic surface is also incompressible. So we can see that, for a given complete finite-area hyperbolic surface, all the subsurfaces with totally geodesic boundary is incompressible. So you can ask, what is a subsurface that is not incompressible? Let's choose S' by removing a disk from a surface S. Then S' is a subsurface of S, and S' has a boundary, which is the boundary of the removed disc. Now let take a loop on S' which is homotopic to the boundary of S'. Hence this loop is essential on S' but it is not essential on S.


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