Laminations and 2-filling rays on infinite type surfaces

TL;DR

This paper constructs and analyzes 2-filling rays on infinite type surfaces, advancing understanding of the boundary of the loop graph and introducing new methods for their classification.

Contribution

It provides the first examples of 2-filling rays on infinite type surfaces and develops multiple construction techniques including combinatorial and train track approaches.

Findings

Constructed the first examples of 2-filling rays on infinite type surfaces.

Described all 2-filling rays with certain properties and their mapping class group orbits.

Connected 2-filling rays to boundary points of the loop graph.

Abstract

The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph as a space of geodesic laminations. We approach this problem by constructing the first examples of 2-filling rays on infinite type surfaces. Such rays accumulate onto geodesic laminations which are in some sense filling, but without strong enough properties to correspond to points in the boundary of the loop graph. We give multiple constructions using both a hands-on combinatorial approach and an approach using train tracks and automorphisms of flat surfaces. In addition, our approaches are sufficiently robust to describe all 2-filling rays with certain other basic properties as well as to produce uncountably many distinct mapping class group orbits.