2024/08/13 by Brandis Whitfield, Whitfield, Brandis · 1 voice · 3 citations
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.2408.07044
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 Mf, its associated mapping torus, has a compactification as a 3-manifold with boundary; further, if f is atoroidal, then Mf admits a hyperbolic metric. Such maps admit invariant positive and negative Handel-Miller laminations, Λ+, Λ-, whose leaves naturally project to the arc and curve complex of a given compact subsurface Y⊂ S. As an end-periodic analogy to work of Minsky in the finite-type setting, we show that for every ε>0 there exists K> 0 (depending only on ε and the capacity of f) for which dY (Λ+, Λ-)≥ K implies infσ∈ AH(Mf)\ℓσ(∂ Y)\ ≤ ε. Here ℓσ(∂ Y) denotes the total geodesic length of ∂ Y in (Mf, σ), and the infimum is taken over all hyperbolic structures on Mf. This work produces the following: given a closed surface Σ, we provide a family of closed, fibered hyperbolic manifolds in which Σ is totally geodesically embedded, (almost) transverse to the pseudo-Anosov flow, with arbitrarily small systole.