On pseudo-Anosov maps, symplectic, Perron-Frobenius matrices, and compression bodies

In 1988, William Thurston announced the completion of a classification of surface automorphisms into three types up to isotopy: periodic, reducible, and pseudo-Anosov. The most common but also least understood maps in this classification are pseudo-Anosovs. We extend our understanding of pseudo-Anosov maps in two ways. First, we show that every Perron unit of appropriate degree has a power which appears as the spectral radius of a symplectic, Perron-Frobenius matrix. This is significant due to possible applications to understanding the spectrum of dilatations for a surface. Second, we present an alternative proof to an important result of Biringer, Johnson, and Minsky showing roughly that a power of a pseudo-Anosov extends over a compression body if and only if the stable lamination bounds. Our alternative proof follows ideas of Casson and Long first presented in 1985