Generalized Fermat Riemann surfaces of infinite type
Ruben A. Hidalgo
TL;DR
This work constructs and analyzes Riemann surface structures on the Loch Ness Monster (LNM) that admit an abelian automorphism group $H \cong {\mathbb Z}_{k}^{\mathbb N}$ with $S/H$ planar, establishing a broad infinite-type analogue of generalized Fermat curves. It develops an inverse-limit, algebraic model $C^{k}_{\infty}(\mathcal{B};M) \subset {\mathbb P}^{\mathbb N}$, together with a Galois cover $\pi_{\infty}$ and an action by $H^{\infty}_{k} \cong {\mathbb Z}_{k}^{\mathbb N}$, thereby producing ${\mathbb Z}_{k}^{\infty}$-gonal surfaces homeomorphic to the LNM. A complementary Fuchsian-uniformization framework connects these infinite-type objects to standard finite-type Fermat curves via subgroups and inverse limits, and reveals a fiber-product perspective tying the infinite-type models to finite-type data. The paper also treats hyperelliptic infinite-type surfaces, giving uniformization and algebraic descriptions, and identifies when such hyperelliptic structures yield the LNM, providing explicit affine models in the finite-${\mathcal B}'$ case. Overall, the results unify topology (LNM), infinite-type algebraic geometry, and Fuchsian group theory to describe a rich, computable class of infinite-genus surfaces with controlled automorphism actions and branched coverings.
Abstract
The Loch Ness monster (LNM) is, up to homeomorphisms, the unique orientable, connected, Hausdorff, second countable surface of infinite genus and with exactly one end. For each integer $k \geq 2$, we construct Riemann surface structures $S$ on the LNM admitting a group of conformal automorphisms $H \cong {\mathbb Z}_{k}^{\mathbb N}$ such that $S/H$ is planar. These structures can be described algebraically inside the projective space ${\mathbb P}^{\mathbb N}$ after deleting some limit points.