Lehmer's number is not an orientable stretch factor in genus six

Abstract

Let $λ_L \approx 1.17628$ be Lehmer's number. We prove that no pseudo-Anosov homeomorphism of a closed orientable surface of genus six with orientable invariant foliations has stretch factor $λ_L$. This gives a negative answer to a question of Lanneau and Thiffeault on the minimal stretch factor in even genus. As a consequence, the minimal stretch factor $δ_6^{+}$ of such maps is the largest root of $x^{12}-x^{9}-x^{7}+x^{6}-x^{5}-x^{3}+1$, approximately $1.20189$. The proof combines the Lefschetz method of Lanneau and Thiffeault, the standardly embedded train tracks of Hironaka and Tsang, an elementary analysis of the transition digraph, and an obstruction from periodic conjugacy classes in the fundamental group.