REPRESENTATIONS OF TRIANGLE GROUPS AND REGULAR MAPS

Mária Ipolyiová

   A regular map is a cellularly embedded graph in an orientable surface whose automorphism group acts regularly on the dart set of the map. If a regular map is of type {m,n} (that is, of face length m and vertex valence n) then its automorphism group is a quotient of a triangle group T(2,m,n) = < x,y; xⁿ = y² = (xy1)^m = 1 > . The groups T(2,m,n) have faithful representations in linear groups SL₃(Z[η,ξ]) where η = 2 cos(π/m) and ξ = 2 cos(π/n). We will investigate the minimal polynomials for η and ξ over Z . In a follow-up paper the results will be applied to constructions of regular maps with large planar width.

Keywords: regular map, triangle group, minimal polynomial

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