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Reflexible regular maps with no non-trivial exponents from residual finiteness

Siran, Jozef; Stanekova, Lubica and Olejar, Marian (2011). Reflexible regular maps with no non-trivial exponents from residual finiteness. Glasgow Mathematical Journal, 53(3) pp. 437–441.

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DOI (Digital Object Identifier) Link: http://dx.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; xm = y2 = (xy)n = … = 1〉; the parameters m and n are the valence and the covalence of the map, respectively. The element j ∈ Zm* is an exponent of such a map if the assignments x ↦ xj 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
Copyright Holders: 2011 Glasgow Mathematical Journal Trust
ISSN: 0017-0895
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 30421
Depositing User: Jozef Širáň
Date Deposited: 09 Dec 2011 10:40
Last Modified: 03 Dec 2012 15:34
URI: http://oro.open.ac.uk/id/eprint/30421
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