Short curves of end-periodic mapping tori
Year: 2026
Type: article
Abstract: 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 its associated mapping torus π π has a compactification as a 3-manifold with boundary; further, if f is atoroidal then π π admits a hyperbolic metric. Such maps admit invariant positive and negative HandelβMiller laminations , Ξ + , Ξ β , whose leaves naturally project to the arc and curve complex of a compact subsurface π β π . As an end-periodic analogy to work of Minsky in the finite-type setting, we show that there exists a hyperbolic structure s of π π such that for every π > 0 there exists πΎ > 0 (depending only on Ξ΅ and the capacity of f ) for which π π β‘ ( Ξ + , Ξ β ) β₯ πΎ implies β π β‘ ( β π ) β€ π . Here β π β‘ ( β π ) denotes the total length of the geodesic representative of β Y in ( π π , π ) . This work additionally produces the following: for every π > 0 and closed connected surface Ξ£ of genus π β₯ 2 , we provide a closed fibered hyperbolic 3-manifold in which Ξ£ embeds as a totally geodesic surface whose systole length is at most Ξ΅ .
Source: Advances in Mathematics
Author: Brandis Whitfield
Institution: University of WisconsinβMadison
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,790
OpenAlex ID: W4402427441