Generalized Fermat Riemann surfaces of infinite type

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.