Universal q-gonal tessellations

Universal q-gonal tessellations

The universal triangular map is the Farey map M^3 as shown by David Singerman in

1988. The orientation-preserving automorphism group of M^3 is the classical modular group 􀀀= PSL(2;Z) and M^3 is universal in the sense that every triangular map on an orientable surface is a quotient of M^3 by a subgroup of 􀀀. In this thesis we describe tessellations M^q of the upper-half complex plane H which are universal for q-gonal maps.

The orientation-preserving automorphism group of M^q is the Hecke groups Hq and we show that every q-gonal map on an orientable surface is of the form M^q=H where H is a subgroup of Hecke groups Hq.

Chapter 1 is devoted to a brief outline of map theory. We define algebraic maps and topological maps, explaining the connections between them.

Chapter 2 is devoted to discussing the modular group and Hecke groups, and describing their fundamental regions.

Chapter 3 is devoted to describing the Farey map and the universal q-gonal tessellations

M^q and showing that M^q is universal, in the sense that every q-gonal map on an orientable surface is a quotient of M^q by a subgroup of Hq.

Chapter 4 is devoted to discussing the principal congruence subgroups of the Hecke groups Hq and the quotients of M^q by these subgroups. An important result gives the index of these subgroups in the Hecke groups in the cases q = 4 and 6, a result given previously by Parsons with a different proof. We then discuss many of these maps for

q = 4 and 6, and also study the combinatorics and geometry of these maps, including the graphical distance, diameter, stars and poles particularly nice example is a quotient of M^4 corresponding to Bring’s curve.

Chapter 5 is devoted to considering the Petrie paths for M^q. These project to Petrie polygons on the quotient maps and we relate the sizes of the Petrie polygons on these maps to the period of the Hecke-Fibonacci sequence modulo n.

University of Southampton

Kattan, Doha A.

1c01375a-e2b3-4a88-b9d1-7cc2474f0bde

July 2019

Kattan, Doha A.

1c01375a-e2b3-4a88-b9d1-7cc2474f0bde

Theriault, Stephen

5e442ce4-8941-41b3-95f1-5e7562fdef80

Kattan, Doha A.
(2019)
Universal q-gonal tessellations.
*University of Southampton, Doctoral Thesis*, 111pp.

Record type:
Thesis
(Doctoral)

## Abstract

The universal triangular map is the Farey map M^3 as shown by David Singerman in

1988. The orientation-preserving automorphism group of M^3 is the classical modular group 􀀀= PSL(2;Z) and M^3 is universal in the sense that every triangular map on an orientable surface is a quotient of M^3 by a subgroup of 􀀀. In this thesis we describe tessellations M^q of the upper-half complex plane H which are universal for q-gonal maps.

The orientation-preserving automorphism group of M^q is the Hecke groups Hq and we show that every q-gonal map on an orientable surface is of the form M^q=H where H is a subgroup of Hecke groups Hq.

Chapter 1 is devoted to a brief outline of map theory. We define algebraic maps and topological maps, explaining the connections between them.

Chapter 2 is devoted to discussing the modular group and Hecke groups, and describing their fundamental regions.

Chapter 3 is devoted to describing the Farey map and the universal q-gonal tessellations

M^q and showing that M^q is universal, in the sense that every q-gonal map on an orientable surface is a quotient of M^q by a subgroup of Hq.

Chapter 4 is devoted to discussing the principal congruence subgroups of the Hecke groups Hq and the quotients of M^q by these subgroups. An important result gives the index of these subgroups in the Hecke groups in the cases q = 4 and 6, a result given previously by Parsons with a different proof. We then discuss many of these maps for

q = 4 and 6, and also study the combinatorics and geometry of these maps, including the graphical distance, diameter, stars and poles particularly nice example is a quotient of M^4 corresponding to Bring’s curve.

Chapter 5 is devoted to considering the Petrie paths for M^q. These project to Petrie polygons on the quotient maps and we relate the sizes of the Petrie polygons on these maps to the period of the Hecke-Fibonacci sequence modulo n.

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## More information

Published date: July 2019

## Identifiers

Local EPrints ID: 451372

URI: http://eprints.soton.ac.uk/id/eprint/451372

PURE UUID: 47bdfed0-d0e2-4488-80e2-f513cdc0fc9c

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Date deposited: 22 Sep 2021 16:32

Last modified: 17 Mar 2024 06:24

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## Contributors

Author:
Doha A. Kattan

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