My approach to Riemann surfaces is via the Fenchel-Nielsen coordinates of the Teichmüller space associated to the unique geodesic pants decomposition of a surface. A pair of pants is homeomorphic to a sphere minus three open disks with Jordan curve boundaries. If a pair of pants is straightened to have closed geodesics as boundary components, it becomes a geodesic pair of pants. Closed geodesics drawn on a surface that separate it into geodesic pairs of pants gives a geodesic pants decomposition. The Fenchel-Nielsen coordinates on the Teichmüller space T(S) are associated to a geodesic pants decomposition of the surface S. Each closed geodesic in the decomposition is associated with its hyperbolic length, called the length coordinate. We also associate to it a twist, or the relative hyperbolic distance between feet of orthogeodesics drawn from the closed geodesic to the nearest closed geodesics on both sides. The lengths and twists constitute the Fenchel-Nielsen coordinates of a surface.

Figure 1: A half-twist tight flute surface X({l_{n}, 1/2}).

Basmajian, Hakobyan, and Šarić in [BHS] extensively studied surfaces called tight flute surfaces. They constructed them in the following way. Start with a geodesic pair of pants with two punctures and a closed geodesic boundary. Attach geodesic pairs of pants with two closed geodesic boundaries and one puncture in an infinite chain. The result is a tight flute surface. They called the closed geodesics between pants, cuffs. Their lengths give the length coordinates {l

_{n}}, and their twists give the twist coordinates {t_{n}}. When the twist of each cuff is one half, you get a half-twist tight flute surface (see Figure 1).

Figure 2: The tail of a half-twist Loch-Ness monster.

In [BHS], the parabolicity of surfaces called Loch-Ness monsters were determined using their Fenchel-Nielsen coordinates. They constructed a Loch-Ness monster by replacing each puncture of a tight flute surface with a closed geodesic, called ß

_{n}, and attaching a torus with a hole to each one. Šarić and I add an additional simple closed geodesic γ_{n}on each attached torus separated by ß_{n}. We assume the lengths of ß_{n}and γ_{n}are between two positive constants. When the twist of every cuff between pairs of pants is one half, the result is a a half-twist Loch-Ness monster. See Figure 2 for a picture of the tail of a half-twist Loch-Ness monster.

Figure 3: A surface with finitely many symmetric ends.

This led in [BHS] to the classification of parabolicity of a more general type of surface whose construction follows. Starting with a surface X

_{0}with closed geodesic boundaries { δ_{1},...,δ_{k}}, attach to each δ_{i}a symmetric non-planar surface with one boundary a closed geodesic. The result is a surface with finitely many symmetric ends, or a symmetric end surface (see Figure 3). Each symmetric end is the tail of a Loch-Ness monster. We call the symmetric end surface minus the attached toruses its body.