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Is anything known about minimum dilatation pseudo-anosovs on non-orientable surfaces?

More specifically it is known whether the asymptotic behavior for log(minimal dilatation) is 1/genus? (The lower bound follows from the lower bound for pseudo-anosovs on orientable surfaces, but I do not know of any construction that realizes it.)

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    $\begingroup$ I believe one ought to be able to construct examples with that asymptotics (although maybe not for all genus) in the same fashion as for the orientable case. Take a non-orientable fibered manifold with $b_1 >1$. The Thurston norm is defined on $H^1$, and one may take a sequence approaching an irrational point on a fibered face of the unit ball. Then McMullen's formula for the dilatation and the linearity of the Thurston norm should imply the same asymptotics. $\endgroup$
    – Ian Agol
    Commented Aug 9, 2013 at 5:31
  • $\begingroup$ @IanAgol Thank you. This seems to answer my question once I convince myself that McMullen's formula works in the non-orientable case as well. Thurston mentions in his paper that most things about the Thurston norm carry to the non-orientable case which I can imagine, but McMullen's proof is tougher... $\endgroup$
    – Balazs Strenner
    Commented Aug 10, 2013 at 3:12
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    $\begingroup$ I think McMullen's proof works in the non-orientable case as well. Take a pseudo-Anosov, and take a double branched cover to make the foliation orientable. Then the dilatation of the pseudo-Anosov is the maximal eigenvalue of the action on homology of this cover. McMullen points out that the eigenvalue is the maximal root of an Alexander polynomial, and for a fixed fibered face, there is a multi-variable Alexander poly. which specializes to the Alexander polys. of each cohomology class. I think all this goes through in the non-orientable case. $\endgroup$
    – Ian Agol
    Commented Aug 10, 2013 at 3:30

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One may construct upper bounds in the non-orientable case the same way as McMullen does in his paper (see p. 523 for a short description of his "renormalization" procedure, or Section 10).

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