Let $S$ be a boundaryless infinite-type surface with finitely many ends and consider an end-periodic homeomorphism $f$ of S. The end-periodicity of $f$ ensures that $M_f$, its associated mapping torus, has a compactification as a $3$-manifold with boundary; and further, if $f$ is atoroidal, then $M_f$ admits a hyperbolic metric.
As an end-periodic analogy to work of Minsky in the finite-type setting, we show that given a subsurface $Y\subset S$, the subsurface projections between the ``positive" and ``negative" Handel-Miller laminations provide bounds for the geodesic length of the boundary of $Y$ as it resides in $M_f$.
In this talk, we'll discuss the motivating theory for finite-type surfaces and closed fibered hyperbolic $3$-manifolds, and how these techniques may be used in the infinite-type setting.