Is it possible to give an elementary proof of the aforementioned result? This was proved by Robert Penner (at "A construction of pseudo-Anosov homeomorphisms". Robert C. Penner. Trans. Amer. Math. Soc., 310(1):179–197, 1988) under the techniques of measured-foliations.
Mahan Mj
The answer is given by Penner at the bottom of pg 196 of the paper you cite:
"On each of the projective plane, Klein bottle, and torus-sum-projective-plane there are unique simple curves with special properties."
Thus diffeomorphisms preserve the free homotopy classes of such curves and are therefore reducible (in particular not pseudo Anosov).
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