Growth tightness of surface groups

We prove that fundamental groups of closed oriented surfaces ∑g of genus g ≥ 2 are growth tight with respect to hyperbolic metrics and to the word metric relative to their canonical presentation: this means that the exponential growth rate of π1 (∑g, with respect to these metrics, is always strictly greater than the corresponding growth rate of any of its proper quotients. As an application, we give a new, purely analytic proof of Hopficity of surface groups.