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 transitive 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.