Universal 𝑞-gonal tessellations and their Petrie paths

Abstract

In [Sin88] the second author showed that the Farey map F \mathbb {F} is a model of the universal triangular tessellation. This is a tessellation that covers every other triangular tessellation on an orientable surface. More precisely, the automorphism group of F \mathbb {F} is the classical modular group Γ = P S L ( 2 , Z ) \Gamma =PSL(2,\mathbb {Z}) and every triangular map is the quotient of F \mathbb {F} by a subgroup of Γ \Gamma . The aim of this paper is to describe universal q q -gonal tessellations. Here the modular group will be replaced by Hecke groups.

In [SS16] it was shown that the Petrie paths of the Farey map pass through vertices whose numerators and denominators are Fibonacci numbers. In section 8 we consider Hecke-Fibonacci sequences which arise out of universal q q -gonal tessellations.