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Maps with highest level of symmetry that are even more symmetric than other such maps: regular maps with largest exponent groups. (English) Zbl 1232.05088
Brualdi, Richard A. (ed.) et al., Combinatorics and graphs. Selected papers based on the presentations at the 20th anniversary conference of IPM on combinatorics, Tehran, Iran, May 15–21, 2009. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4865-4/pbk). Contemporary Mathematics 531, 95-102 (2010).
Summary: Regular maps are generalizations of Platonic solids and can be identified with two-generator presentations of groups \(G\) of the form \(\langle x,y; x^2= y^m= (xy)^n=\cdots= 1\rangle\); the parameters \(m\) and \(n\) are the degree and the face length of the map. Such maps have the ‘highest level’ of orientation-preserving symmetry among all maps.
A regular map of vertex degree \(m\) is said to have exponent \(j\in Z^*_m\) if the assignment \(x\mapsto x\) and \(y\mapsto y^j\) extends to an automorphism of \(G\). Any exponent induces an automorphism of the underlying graph which can be viewed as an ‘external symmetry’ of the map. Exponents of a map form a subgroup of \(Z^*_m\) and hence \(Z^*_m\) is the theoretically largest possible group of exponents a regular map of degree \(m\) can have.
In this paper we show that for any given \(m\geq 3\) there exist infinitely many finite regular maps of degree \(m\) with exponent group equal to \(Z^*_m\). We also show that this result does not, in general, extend to regular maps of given degree and given face length.
For the entire collection see [Zbl 1202.05003].

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F05 Generators, relations, and presentations of groups
57M15 Relations of low-dimensional topology with graph theory