On the topology of infinite regular and chiral maps

Abstract

We prove that infinite regular and chiral maps take place on surfaces with at most one end. Moreover, we prove that an infinite regular or chiral map on an orientable surface with genus can only be realized on the Loch Ness monster, that is, the topological surface of infinite genus with one end.Key words and phrases. Infinite surface, regular and chiral polytope. 1 We think of Γ as the geometric realization of an abstract graph.

On the Infinite Loch Ness monster

“…Later they wrote a paper calling this mathematical objects the skew polyhedra [Cox36], or also known today as the Coxeter-Petrie polyhedra. Indeed, they are topologically equivalent to the Infinite Loch Ness monster as shown by the authors jointly with Ferrán Valdez in [ARMV17]. Given that from a combinatorics view, one can think that skew polyhedra are multiple covers of the first three Platonic solids, John H. Conway and et.…”

Geometric Schotkky groups and non compact hyperbolic surface with infinite genus

“…Between this surfaces stand out the Infinite Loch Ness monster (the only surface having infinitely many handles and only one way to go to infinity) and the Jacob's ladder (the only orientable surface having two ways to go to infinity and infinitely many handles in each) see [PS81] and [Ghy95], which are some of the usual examples in this field, in fact, in [ARM17] the authors give an explicit Fuchsian group Γ to generated a Loch Ness monster with hyperbolic structure as quotient H/Γ. Motivated by this particularity, the various investigations on non-compact Riemann surfaces (see e.g., [AMV17], [LT16], [Mat18], [RMV17], and others) and the characterization given by the uniformization theorem in terms of universal covers for Riemann surfaces (see e.g. [Abi81], [FK92]), naturally arises the following inquiry: Question 1.1.…”