Expository Article: Young person's guide to translation surfaces of genus two: McMullen's connected sum theorem

L-cut splitting of translation surfaces and non-embedding of pseudo-Anosovs (in genus two)

“…This induces the "antenna" in the polygon P 2 presenting torus M 2 . Corollary 1.3 is analogous to the result of McMullen in [27] (see also [5]) asserting that any translation surface M can be rotated by some angle θ to become a connected sum of two tori M 1 and M 2 joined along two vertical cuts, J 1 in M 1 and J 2 in M 2 . (Rotating M refers to changing the translation surface structure by postcomposing charts into R 2 with a rotation by θ.)…”

“…All of the above discussion can be repeated for M ∈ H(1, 1) (see e.g. [5]). Roughly, the four verticals incoming into the two 4π singularities z 0 and z 1 cut I into five segments, labeled A, B, C, D, and E (Figure 4.3).…”

“…The central symmetry induces on M an isometric involution η : M → M with six fixed points. This is the Weierstrass involution of M considered as a Riemann surface (see [9] and [5]). Note that η * : H 1 (M, R) → H 1 (M, R) takes any homology class to its opposite, i.e., η * = −Id.…”