Regular maps constructed from linear groups (Q1321678)

The authors use quotients of certain finite linear groups to construct regular maps of surfaces. For an integer \(m > 2\), let \(L_ 2^{\langle - 1 \rangle} (Z_ m)\) be the group of \(2 \times 2\)-matrices over \(Z_ m\) with determinants \(\pm 1\), and let \(C^{\langle - 1 \rangle} (m)\) be its center. For each subgroup \(H\) with \(\{\pm 1\} \leq H \leq C^{\langle - 1 \rangle} (m)\) there is a regular map of type \(\{3, m\}\) whose group is \(L_ 2^{\langle - 1 \rangle} (Z_ m)/H\). The ways in which such maps cover another, or in which they decompose as a blend of simpler maps are determined by the prime factorization of \(m\). Several well-known maps arise in this way. The authors also discuss realizations for the maps based on the plane \(Z_ m^ 2\).