Hyperbolic symmetric rigidity and intrinsic surface geometry

We develop a theory of symmetric infinitesimal rigidity for bar-joint frameworks in the hyperbolic plane, where symmetry is given by a Fuchsian group acting by isometries. Using the language of gain graphs and the orbit rigidity matrix, we reduce rigidity questions for infinite symmetric frameworks in the upper-half plane H to finite combinatorial conditions. Our main result provides a combinatorial characterisation of the infinitesimal rigidity for Gamma-symmetric frameworks, which are as generic as possibly allowed by the symmetry, when Gamma is a surface group. Namely, we show that a Gamma-gain graph is Gamma-isostatic if and only if it satisfies certain matroidal sparsity conditions. In particular, if Gamma is not cyclic then the appropriate combinatorial condition is (2,3,1,0)-gain tightness. Via the correspondence between Gamma-symmetric frameworks in H and finite frameworks on the quotient surface H/Gamma, this yields a characterisation of infinitesimal rigidity for frameworks on compact Riemann surfaces of genus at least 2.