There are no exotic ladder surfaces

Abstract

It is an open problem to provide a characterization of quasiconformally homogeneous Riemann surfaces. We show that given the current literature, this problem can be broken into four open cases with respect to the topology of the underlying surface. The main result is a characterization in one of the these open cases; in particular, we prove that every quasiconformally homogeneous ladder surface is quasiconformally equivalent to a regular cover of a closed surface (or, in other words, there are no exotic ladder surfaces).

Figures (4)

• Figure 1: On the left, $P$ is a hyperbolic right-angled pentagon and four copies of $P$ are glued to form a square with a disk removed. On the right, these 1-holed squares are glued in a tiling extending in all directions; the extension of the vertical geodesic $\alpha$ forms a proper geodesic arc.
• Figure 2: A pants decomposition for a (topological) ladder surface.
• Figure 3: The elementary move up to homeomorphism in a once-punctured torus switching $b$ and $b'$
• Figure 5: The pants decomposition $\mathcal{P}$ along with the seams $\{d_k\}_{k\in \mathbb Z}$ determine Fenchel-Nielsen coordinates for hyperbolic structures on $S$.

Theorems & Definitions (39)

• Corollary 1.1
• Corollary 1.2
• Lemma 2.1
• Theorem 2.2
• Theorem 2.3: Bers pants decomposition theorem
• Theorem 2.4: Classification of surfaces (see RichardsClassification)
• Lemma 3.1
• proof
• Theorem 3.2
• Lemma 3.3