Closed surface groups are hyperbolic

By [L4], the intrinsic curvature hypothesis gives an isometric identification of the universal cover with ${\mathbb{H}}^2$. Under it, ${\pi }_1(\mathrm{\Sigma })$ acts isometrically, properly and cocompactly.

By [L2], this action makes ${\pi }_1(\mathrm{\Sigma })$ finitely generated and its word-metric vertex set quasi-isometric to ${\mathbb{H}}^2$. For the finite generating set supplied there, join each edge's orbit endpoints by a chosen geodesic in ${\mathbb{H}}^2$ and extend the orbit map over that unit edge. The finitely many generator displacements give a uniform edge-image diameter. Every point of the geometric Cayley graph is within $1\mathrm{/}2$ of a vertex, so the vertex quasi-isometry inequalities extend to the whole graph with only bounded additive errors, and coarse density is unchanged. Thus the two geodesic spaces are quasi-isometric. By [L1] the plane is hyperbolic; [L3] makes this geometric Cayley graph hyperbolic.