L-cuts for genus two translation surfaces
Bouwman, Andrew Kevin.
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A connected sum is a topological way of joining two Riemann surfaces which results in another surface. It is used in the classification of all connected closed orientable surfaces as being homeomorphic to either the sphere, or a connected sum of tori. The reverse operation, here referred to as a splitting, decomposes a surface as a connected sum. It was recently shown by Curtis McMullen that any translation surface of genus two can be written in infinitely many ways as a connected sum of two flat tori. His method was to find a certain straight saddle connection J, and perform a splitting along J U eta(J), where eta is the hyperelliptic involution (the unique degree-two automorphism on the surface which fixes exactly six points). In this dissertation, we give an elementary argument for existence of such J and show that for all surfaces of genus two on which the vertical flow is minimal, the same kind of splitting is possible along a parallel pair of paths with the straight saddle connection replaced by an L-cut: a broken line with one horizontal and one vertical segment. As a direct consequence of this L-cut splitting, it is shown that a homeomorphism on a genus-two surface which is conjugated to a hyperbolic toral automorphism restricted to an invariant subset (if any such situation even exists) can only be pseudo-Anosov with non-orientable foliations. This makes progress toward addressing an old question of Stephen Smale about the existence of an invariant set of a hyperbolic toral automorphism which is itself a compact surface.