Abstract: A theorem of Nielsen-Thurston tells us that every mapping class of a closed surface (i.e. a homotopy class of homeomorphisms) is one of three types: periodic, reducible or pseudo-Anosov. Pseudo-Anosov mapping classes are precisely those with a representative preserving a pair of transverse (measured) foliations and they have shown to be generic in the mapping class group due mainly to work of Maher and Rivin. One of the main ways to construct explicit pseudo-Anosov mapping classes is via the Penner-Thurston construction, and fairly recently, it was shown by Shin—Strenner that not all pseudo-Anosov mapping classes arise from this construction. In this talk, I will discuss how dense/generic the Penner-Thurston pseudo-Anosovs are in some subsemi-groups of the mapping class group. Joint work with Josh Pankau and Jing Tao.