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Universal q-gonal tessellations and their Petrie paths

Universal q-gonal tessellations and their Petrie paths
Universal q-gonal tessellations and their Petrie paths

In [Sin88] the second author showed that the Farey map 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 is the classical modular group Γ = PSL(2, Z) and every triangular map is the quotient of F by a subgroup of Γ. The aim of this paper is to describe universal 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-gonal tessellations.

Farey map, Hecke groups, Hecke-Fibonacci sequence, Map, Petrie path, Universal tessellation
0271-4132
American Mathematical Society
Kattan, Doha
1c01375a-e2b3-4a88-b9d1-7cc2474f0bde
Singerman, David
3eeb0783-c87c-4405-81d7-e80ae4c15f8b
Wootton, Aaron
Broughton, S. Allen
Paulhus, Jennifer
Kattan, Doha
1c01375a-e2b3-4a88-b9d1-7cc2474f0bde
Singerman, David
3eeb0783-c87c-4405-81d7-e80ae4c15f8b
Wootton, Aaron
Broughton, S. Allen
Paulhus, Jennifer

Kattan, Doha and Singerman, David (2022) Universal q-gonal tessellations and their Petrie paths. In, Wootton, Aaron, Broughton, S. Allen and Paulhus, Jennifer (eds.) Automorphisms of Riemann Surfaces, Subgroups of Mapping Class Groups and Related Topics. (Contemporary Mathematics, 776) American Mathematical Society. (doi:10.1090/conm/776/15617).

Record type: Book Section

Abstract

In [Sin88] the second author showed that the Farey map 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 is the classical modular group Γ = PSL(2, Z) and every triangular map is the quotient of F by a subgroup of Γ. The aim of this paper is to describe universal 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-gonal tessellations.

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Published date: 2022
Keywords: Farey map, Hecke groups, Hecke-Fibonacci sequence, Map, Petrie path, Universal tessellation

Identifiers

Local EPrints ID: 457863
URI: http://eprints.soton.ac.uk/id/eprint/457863
ISSN: 0271-4132
PURE UUID: e4362000-41a7-419a-999c-29fbb6e1e340

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Date deposited: 21 Jun 2022 17:58
Last modified: 16 Mar 2024 18:03

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Contributors

Author: Doha Kattan
Author: David Singerman
Editor: Aaron Wootton
Editor: S. Allen Broughton
Editor: Jennifer Paulhus

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