Siran, Jozef; Stanekova, Lubica and Olejar, Marian
(2011).
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DOI (Digital Object Identifier) Link: | http://doi.org/10.1017/S0017089511000048 |
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Abstract
Regular maps, that is, graph embeddings with the ‘highest level’ of orientation-preserving symmetry, can be identified with two-generator presentations of groups G of the form 〈x, y; x^{m} = y^{2} = (xy)^{n} = … = 1〉; the parameters m and n are the valence and the covalence of the map, respectively. The element j ∈ Z_{m}^{*} is an exponent of such a map if the assignments x ↦ x^{j} and y ↦ y extend to an automorphism of G. The element −1 is an exponent if and only if the map is reflexible, that is, isomorphic to its mirror image. Non-trivial exponents, j ≠ ±1, induce automorphisms of the underlying graph, which are not map automorphisms but can nevertheless be considered as ‘external symmetries’ of the map. In this paper we show with the help of residual finiteness of triangle groups that for any given m, n≥3 such that 1/m + 1/n ≤ 1/2, there exist infinitely many finite, reflexible and regular maps of valence m and covalence n with no non-trivial exponent.
Item Type: | Journal Article |
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Copyright Holders: | 2011 Glasgow Mathematical Journal Trust |
ISSN: | 0017-0895 |
Academic Unit/Department: | Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics Faculty of Science, Technology, Engineering and Mathematics (STEM) |
Item ID: | 30421 |
Depositing User: | Jozef Širáň |
Date Deposited: | 09 Dec 2011 10:40 |
Last Modified: | 06 Aug 2016 05:27 |
URI: | http://oro.open.ac.uk/id/eprint/30421 |
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