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# Universal q-gonal tessellations

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 &#x100000;= 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 &#x100000;. 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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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
ORCID for Stephen Theriault: orcid.org/0000-0002-7729-5527

## Catalogue record

Date deposited: 22 Sep 2021 16:32

## Contributors

Author: Doha A. Kattan