Polynomial genus dependence for Steiner spanners on constant-curvature surfaces

Develop Steiner (1+ε)-spanners for n points on constant-curvature surfaces whose edge count depends polynomially on the genus g, improving on the current bounds that incur g^{O(g)} factors via Dirichlet-domain representative constructions.

Background

The authors extend their spanner constructions to certain quotient spaces and closed orientable constant-curvature surfaces, but the present approach introduces a g^{O(g)} multiplicative blow-up due to the number of representative copies needed via Dirichlet domains.

Achieving polynomial dependence on genus would significantly strengthen the practical and theoretical applicability of Steiner spanners on surfaces, aligning the bounds more closely with Euclidean and spherical cases.