Klara Stokes: Rigidity of graphs on compact Riemann surfaces

Abstract: A bar-and-joint framework is a realisation of a graph in a metric space, and it is rigid if the only motions preserving the distances between adjacent vertices are trivial. The rigidity of frameworks in the Euclidean plane can be determined using combinatorics only, and the intent to generalise this theory to higher dimensions has provoked quite a lot of activity, in particular in matroid theory. In this talk I will consider a framework moving on a compact Riemann surface of constant curvature, with its edges geodesics whose lengths are preserved (rather than treating the bars as straight in an ambient space). While frameworks on the sphere and torus are well understood, the rigidity problem was unsolved for compact surfaces of higher genus. By uniformisation, such a surface is a quotient of the hyperbolic plane by a surface group \(S < \mathrm{PSL}(2,\mathbb{R})\), so the framework lifts to an \(S\)-symmetric framework in the hyperbolic plane, encoded by an \(S\)-gain graph. I will explain how rigidity is then governed by the subgroups of \(S\) generated by the gains of cycles. Depending on whether such a subgroup is trivial, cyclic or non-cyclic, its centraliser in \(\mathrm{PSL}(2,\mathbb{R})\) has dimension 3, 1 or 0, and this dimension determines the count. Together with the subgroup structure of surface groups, this leads to a combinatorial counting characterisation of generic rigidity. This is joint work with Sean Dewar, Alison La Porta, Rebecca Monks, Anthony Nixon and Joannes Vermant.