nLab cartographic group

The group

𝒞 2=σ 0,σ 1,σ 2σ 0 2=σ 1 2=σ 2 2=(σ 0σ 2) 2=1\mathcal{C}_2 = \langle\sigma_0,\sigma_1,\sigma_2 \mid \sigma_0^2 = \sigma_1^2 = \sigma_2^2 = (\sigma_0\sigma_2)^2 = 1\rangle

is called the cartographic group (of dimension 2), while its index 2 subgroup

𝒞 2 +=ρ 0,ρ 1,ρ 2ρ 1 2=ρ 0ρ 1ρ 2=1\mathcal{C}_2^+ = \langle\rho_0,\rho_1,\rho_2 \mid \rho_1^2 = \rho_0\rho_1\rho_2 = 1\rangle

is called the oriented cartographic group. Specifically, this terminology comes from Grothendieck‘s Esquisse d'un programme, and is motivated by the fact that transitive permutation representations (or equivalently, conjugacy classes of subgroups) of 𝒞 2 +\mathcal{C}_2^+ can be identified with topological maps on connected, oriented surfaces without boundary, while more generally, transitive permutation representations of 𝒞 2\mathcal{C}_2 can be identified with maps on connected surfaces which may or may not be orientable or have a boundary.

Higher dimensions

The nn-dimensional analogue of the cartographic group is

𝒞 n=σ 0,,σ nσ i 2=(σ iσ j) 2=1,(|ij|>1),\mathcal{C}_n = \langle\sigma_0,\dots,\sigma_n \mid \sigma_i^2 = (\sigma_i\sigma_j)^2 = 1, (|i-j|\gt 1)\rangle,

which is a Coxeter group. For related references, see the last section of Jones and Singerman.


  • Alexander Grothendieck. Esquisse d'un programme (section 3)
  • Christine Voisin? and Jean Malgoire?. Cartes cellulaires, Cahiers Mathématiques, 12, Montpellier, 1977. (sudoc)
  • Gareth A. Jones and David Singerman. Theory of Maps on Orientable Surfaces. Proceedings of the London Mathematical Society, 37:273-307, 1978. (doi)
  • Robin P. Bryant and David Singerman. Foundation of the Theory of Maps on Surfaces With Boundary. Quarterly Journal of Mathematics, 2(36):17-41, 1985. (doi)
  • Gareth Jones and David Singerman. Maps, hypermaps, and triangle groups. In The Grothendieck Theory of Dessins d’Enfants, L. Schneps (ed.), London Mathematical Society Lecture Note Series 200, Cambridge University Press, 1994. (doi)
  • See also the Wikipedia page on generalized maps, which correspond to permutation representations of 𝒞 n\mathcal{C}_n.

Last revised on September 26, 2023 at 19:59:23. See the history of this page for a list of all contributions to it.