Pseudo-Anosov homeomorphisms not arising from branched covers (Q2180199)

The authors prove that, for any even number \(d\geq 4\) and any \(g\geq \frac{d}{2}+2\), there exists a pseudo-Anosov map \(f:S_g\to S_g\) with orientable invariant stable/unstable foliation, such that the degree of the dilatation \(\lambda(f)\) over \(\mathbb{Q}\) is \(d\), and \(f\) is not a lift of a pseudo-Anosov map on another surface \(\Sigma\) (with smaller genus) via a branched cover \(S_g\to \Sigma\) of degree at least \(2\). In [Proc. Am. Math. Soc. 127, No. 7, 2183--2192 (1999; Zbl 0920.58035)], \textit{J. Franks} and \textit{E. Rykken} proved that, for any pseudo-Anosov map \(f:S\to S\) with orientable invariant stable/unstable foliation whose dilatation is a quadratic irrational, \(f\) is a lift of an Anosov map on the torus \(T^2\) via a branched cover \(S\to T^2\). Then B. Farb (see [\textit{B. Strenner}, Geom. Funct. Anal. 27, No. 6, 1497--1539 (2017; Zbl 1382.37046)]) asked whether this phenomenon holds in general, that is, whether a pseudo-Anosov map with low algebraic degree dilatation over \(\mathbb{Q}\) is a lift of a pseudo-Anosov map on a low genus surface via a branched cover of degree at least \(2\). The authors give a negative answer to Farb's question in this paper. The pseudo-Anosov maps constructed in this paper are given by \textit{R. C. Penner}'s construction [Trans. Am. Math. Soc. 310, No. 1, 179--197 (1988; Zbl 0706.57008)]. For \(r=\frac{d}{2}\) and any \(g\geq r+2\), the authors carefully construct two multicurves \(\{a_1,\dots,a_r\}\) and \(\{b_1,\dots,b_r\}\) that fill the closed orientable surface \(S_g\) with genus \(g\). For each \(m>0\), they carefully take a pseudo-Anosov mapping classe \(f_m\) in the semigroup generated by \(\{T_{a_1}^m,\dots, T_{a_r}^m,T_{b_1}^{-m},\dots, T_{b_r}^{-m}\}\). Then they use \textit{B. Strenner}'s work [Geom. Funct. Anal. 27, No. 6, 1497--1539 (2017; Zbl 1382.37046)] to prove that the dilatation of each \(f_m\) has degree \(2r\) over \(\mathbb{Q}\). To prove that \(f_m\) is not a lift via a branched cover of degree at least \(2\), the authors actually prove that the induced singular flat metric on \(S_g\) is not a lift of a singular flat metric on another surface \(\Sigma\) via a branched cover \(S_g\to \Sigma\) of degree at least \(2\). More precisely, by invoking Mazur-Minsky's machinery on subsurface projection of curve complexes, the authors study extensively moduli of flat cylinders in \(S_g\) and \(\Sigma\). They prove that, for large enough \(m\), the branched cover \(S_g\to \Sigma\), if exists, must have degree \(2\) and the deck transformation on \(S_g\) must be an involution that fixes each \(a_i\) and \(b_i\) curve. However, by the construction of \(f_m\), the combinatorics of \(\{a_1,\dots,a_r\}\) and \(\{b_1,\dots,b_r\}\) on \(S_g\) prohibits the existence of such an involution. Since the authors' proof proves that the induced singular flat metric on \(S_g\) is not a lift via a branched cover, they actually prove that any pseudo-Anosov element in the Veech group that preserves this singular flat metric is not a lift. However, it is possible that this Veech group only consists of powers of \(f_m\). In the end of the paper, by using Thurston's construction, the authors also construct examples satisfying the main theorem with \(d=4\) such that the corresponding Veech group is not elementary.