Abstract: The Mucube and Mutetrahedron are triply periodic polyhedral surfaces that are infinite covers of compact half-translation surfaces. Unlike compact translation surfaces—which exhibit quadratic cylinder growth—we show that the number of isotopy classes of cylinders of length at most $R$ grows subquadratically and at least linearly in $R.$ We establish this by applying a recent Veech group computation for the Mucube and introducing a homological characterization of cylinder curves for the Mutetrahedron, which also yields density of its cylinder directions.
Mathematics Seminars and Colloquia