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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.
Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-g
Let $F_g$ be a closed orientable surface of genus $g$. A set $Omega = { gamma_1, dots, gamma_s}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a emph{filling system} or simply a emph{filling} of $F_g$, if $F_gsetminus Omega$ is a
We prove a quantitative version of the non-uniform hyperbolicity of the Teichmuller geodesic flow. Namely, at each point of any Teichmuller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the flow line in ter
We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results
This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $varepsilon$-fills the surface.