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Distinctness of Two Pseudo-Anosov Homeomorphisms
Kidder 274
Ph.D. DefenseSpeaker: Mesa Walker
In 1981, Arnoux and Yoccoz constructed the first known family of odd degree pseudo-Anosov homeomorphisms. In 1985, David Fried constructed another family of pseudo-Anosov homeomorphisms using completely different methods. Fried’s genus 3 homeomorphism and Arnoux and Yoccoz’s genus 3 homeomorphism both have the same stretch factor, and Fried asked if these maps are distinct or the same. We show that these maps are distinct. We prove the distinctness by blowing up Arnoux and Yoccoz's surface at its cone singularities and applying Fried's method of studying cross sections of mapping tori. We find in our analysis that there is no torus with two points blown up as a cross section of the mapping torus of Arnoux and Yoccoz's blown up homeomorphism. However, Fried's example similarly treated must lead to such a cross section. We can therefore conclude that the pseudo-Anosov homeomorphism found by Arnoux and Yoccoz is distinct from the one found by Fried, as the mapping tori are distinct. Read more.