The universal cover of a closed hyperbolic surface is the hyperbolic plane with geometric deck action

A smooth surface is locally path connected and semilocally simply connected by coordinate disks. By [F1] choose its universal cover $p:\tilde{\mathrm{\Sigma }}\to \mathrm{\Sigma }$. Pull back the smooth coordinate charts and metric along the evenly covered sheets. Then $p$ is a smooth local isometry; in particular the cover has curvature $-1$ and is simply connected. The base $\mathrm{\Sigma }$ is compact, so every geodesic in it extends for all real time by [F2]. A geodesic in $\tilde{\mathrm{\Sigma }}$ projects locally to one in $\mathrm{\Sigma }$; extend the projected geodesic and lift its extended path from the starting point. Uniqueness of path lifting and the local-isometry equation show that the lift extends the original geodesic. Hence $\tilde{\mathrm{\Sigma }}$ is geodesically complete and, by [F2], metrically complete.

Fix $\tilde{p}\in \tilde{\mathrm{\Sigma }}$ and an orthonormal oriented basis of its tangent plane. Completeness defines ${\exp }_{\tilde{p}}$ on every tangent vector. Along any unit-speed radial geodesic ${\gamma }_v(t)={\exp }_{\tilde{p}}(tv)$, a normal variation of its initial direction gives a Jacobi field $J$ with $J(0)=0$ and $D_{t}J(0)=E_0$, where $E_0$ is a unit normal vector. The geodesic-variation equation $D_t^{2}J+R(J,\dot{\gamma })\dot{\gamma }=0$ follows by differentiating the geodesic equation with respect to the variation parameter and commuting the two covariant derivatives. Since the sectional curvature is $-1$, $R(J,\dot{\gamma })\dot{\gamma }=-J$ for perpendicular $J$. Parallel-transport $E_0$ along $\gamma$; uniqueness of the scalar ODE $j^{\mathrm{\prime }\mathrm{\prime }}-j=0$, $j(0)=0$, $j^{\mathrm{\prime }}(0)=1$, gives $J(t)=\sinh (t)E(t)$. Radial variation gives $D\exp$ of norm $1$ in the radial direction, and Gauss' lemma (orthogonality follows by differentiating $⟨\dot{\gamma },J⟩$ and using those initial conditions) gives zero cross term. Thus the pulled-back metric on the whole tangent plane, away from its origin, is ${\exp }_{\tilde{p}}^{\ast }g=d{r}^2+{\sinh }^2(r)d{\theta }^2\mathrm{.}$ At the origin its differential is the identity. Because $\sinh (r)>0$ for $r>0$, ${\exp }_{\tilde{p}}$ is a local diffeomorphism everywhere. This is the constant-curvature calculation in Datar, §24.1, with $\kappa =-1$; all its local ODE steps have been displayed here.

For clarity, the local diffeomorphism in step 1.2 is a covering map. In the Euclidean norm on $T_{\tilde{p}}\tilde{\mathrm{\Sigma }}$, its differential expands every tangent vector: radial length is unchanged and angular length is multiplied by $\sinh (r)\mathrm{/}r\ge 1$ (the inequality follows from the positive Taylor series, or from $(\sinh r-r{)}^{\mathrm{\prime }}=\cosh r-1\ge 0$). A locally lifted piecewise smooth path $c:[0,1]\to \tilde{\mathrm{\Sigma }}$ therefore has Euclidean lift speed no greater than $\mathrm{\mid }{c}^{\mathrm{\prime }}{\mathrm{\mid }}_g$. On any unfinished finite subinterval the lift stays in a bounded Euclidean ball and is Cauchy as the parameter approaches its endpoint; it extends there by continuity of $\exp$ and a local inverse. Hence every such path lifts to its full interval. The same estimate, uniform over a compact parameter square after subdividing it into finitely many normal-coordinate rectangles, lifts piecewise smooth path homotopies. Given a sufficiently small simply connected normal ball $B$ around a target point, lift its radial paths from each point of the fibre. Homotopy lifting makes each lift independent of the path in $B$, producing disjoint inverse branches on $B$; uniqueness of path lifting exhausts its preimage. Therefore $B$ is evenly covered. The target $\tilde{\mathrm{\Sigma }}$ is simply connected, so this connected covering from the tangent plane has one sheet and is a diffeomorphism.

Every deck transformation preserves $p^{\ast }g$, so it is an isometry. The universal covering is regular and [F1] identifies its deck group with ${\pi }_1(\mathrm{\Sigma })$ after a basepoint choice. Its action is free. For compact-set properness, let $K\subseteq \tilde{\mathrm{\Sigma }}$ be compact. Cover $p(K)$ by finitely many smaller disks whose closures lie in evenly covered disks $B_a$. For each $a$, $K$ meets only finitely many sheets above the smaller disk: otherwise points chosen in distinct sheets would accumulate in $K$ over its closure inside $B_a$, where an open sheet contains a neighbourhood of the limit point, a contradiction. If $gK\cap K\mathrm{\ne }\mathrm{\varnothing }$, some $x,gx\in K$ lie over one smaller disk and in two of its finitely many relevant sheets. At most one deck transformation maps one prescribed sheet to the other, so only finitely many $g$ meet $K$. Step 1.1 and [F2] make the cover a proper metric space; arbitrary bounded $B,C$ lie in compact closed balls, and their transporter lies in the finite transporter of the union of those balls. Thus the action is proper in [F3]'s exact bounded-set sense.

The standard polar metric on ${\mathbb{H}}^2$ is $d{r}^2+{\sinh }^2(r)d{\theta }^2$; for example this follows directly from the hyperboloid model $F(r,\theta )=(\cosh r,\sinh r\cos \theta ,\sinh r\sin \theta )$ and the restriction of $-d{x}_0^2+d{x}_1^2+d{x}_2^2$. Match the chosen tangent basis to the corresponding tangent basis at $F(0,\theta )$. The global normal-coordinate diffeomorphism of step 2.1 and the metric identity of step 1.2 now give an isometry $\tilde{\mathrm{\Sigma }}\cong {\mathbb{H}}^2$.

To obtain a bounded set whose translates cover, cover the compact base $\mathrm{\Sigma }$ by finitely many smaller closed normal disks contained in evenly covered disks. Lift each closed disk to one sheet; its lift is compact because the sheet projection is a homeomorphism. Let $K_0$ be the finite union of these lifted compact disks. For any $x\in \tilde{\mathrm{\Sigma }}$, its projection lies in one of them; regularity of the universal cover gives a deck transformation taking the chosen lift over $p(x)$ to $x$. Hence $G\cdot K_0=\tilde{\mathrm{\Sigma }}$. The compact set $K_0$ is bounded, so the action is cobounded by [F3]. Its quotient is the compact surface $\mathrm{\Sigma }$, so it is also cocompact in the usual sense. Together with step 2.2 the action is geometric. The cover is isometric to ${\mathbb{H}}^2$ by step 3.1.