Mehdi Yazdi
Surface homeomorphisms and their lifts by covering maps

A generic surface homeomorphism (up to isotopy) is called 
pseudo-Anosov. These maps come equipped with an algebraic integer that measures 
how much the map stretches/shrinks in different direction, called stretch factor. 
Given a surface homeomorphism, one can ask if it is the lift (by a branched or 
un-branched cover) of another homeomorphism on a simpler surface possibly of small genus. 
Benson Farb conjectured (2004) that if the algebraic degree of the stretch factor is bounded above, then
the map can be obtained by lifting another homeomorphism on a surface of bounded genus. 
This was known to be true for quadratic algebraic integers by a Theorem of Franks-Rykken. 
We construct counterexamples to Farb's conjecture and moreover, provide a lower bound for 
the simplest such cover. The proof is based on an analysis of the Galois conjugates of the 
stretch factor.