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May 8, 2012 at 20:13 comment added Autumn Kent Also, you should get other rigid examples by taking subgroups that are purely filling with respect to a subsurface.
May 8, 2012 at 20:12 comment added Autumn Kent Leininger has pointed out to me that the set of arcs does not depend on $\Gamma$, since they are determined by topological data (the lifts of simple curves on $S$ to $\mathbb{H}^2$ ``link" the limit set, and this linking data, which is really topological, determines the arcs you get). So the map $T(\Gamma) \to M(\Gamma_B)$ will be bounded for these filling groups $B$.
May 6, 2012 at 23:38 vote accept Lee Mosher
May 6, 2012 at 23:24 comment added Lee Mosher Very nice!
May 6, 2012 at 21:32 history answered Autumn Kent CC BY-SA 3.0