To study the relationship between dynamics and structure in a glass-forming liquid, we introduce a purely geometric criterion for locally mobile particles in a dense hard-sphere fluid: namely, "T1-active" particles, which can gain or lose at least one Voronoi neighbor by moving within their free volume with other particles fixed. We obtain geometrical and dynamical properties for monodisperse hard-sphere fluids with 0.40 < ϕ < 0.64 using a "crystal-avoiding" MD simulation that effectively suppresses crystallization without altering the dynamics. We find that the fraction of T1-active particles vanishes at random close packing, while the percolation threshold of T1-inactive particles is essentially identical to the commonly identified hard-sphere glass transition, ϕg ≈ 0.585.