Symmetric isostatic frameworks with ${\mathrm{\ell }}^1$ or ${\mathrm{\ell }}^{\mathrm{\infty }}$ distance constraints

Combinatorial characterisations of minimal rigidity are obtained for symmetric 2-dimensional bar-joint frameworks with either ${\mathrm{\ell }}^1$ or ${\mathrm{\ell }}^{\mathrm{\infty }}$ distance constraints. The characterisations are expressed in terms of symmetric tree packings and the number of edges fixed by the symmetry operations. The proof uses new Henneberg-type inductive construction schemes.