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The loop graph of an infinite type surface is an infinite diameter hyperbolicgraph first studied in detail by Juliette Bavard. An important open problem inthe study of infinite type surfaces is to describe the boundary of the loopgraph as a space of geodesic laminations. We approach this problem byconstructing 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 ofthe loop graph. We give multiple constructions using both a hands-oncombinatorial approach and an approach using train tracks and automorphisms offlat surfaces. In addition, our approaches are sufficiently robust to describeall 2-filling rays with certain other basic properties as well as to produceuncountably many distinct mapping class group orbits.

Laminations and 2-filling rays on infinite type surfaces