Annulus and punctured disc have hyperbolic universal covers

Setup. $S_r$ and $H_{-}$ are open vertical strips and half-planes, hence convex and connected, and $A_r=A(0;r,1)$, ${\mathbb{D}}^{\ast }=A(0;0,1)$ are the annuli of inner radius $r$ and $0$; all four sets are nonempty connected open subsets of $\mathbb{C}$, hence connected Riemann surfaces with the atlas of inclusions.

Injectivity on small discs. If $\mathrm{\mid }w-w^{\mathrm{\prime }}\mathrm{\mid }<2\pi$ and $\exp w=\exp w^{\mathrm{\prime }}$, then $w-w^{\mathrm{\prime }}\in 2\pi i\mathbb{Z}$ [F1], so either $w=w^{\mathrm{\prime }}$ or $\mathrm{\mid }w-w^{\mathrm{\prime }}\mathrm{\mid }\ge 2\pi$, a contradiction; hence $\exp$ is injective on every subset of diameter $<2\pi$, in particular on every disc of radius at most $\pi$.

The principal logarithm. The principal logarithm $\mathrm{Log}$ is a biholomorphism from the slit plane $\mathbb{C}\setminus (-\mathrm{\infty },0]$ onto the horizontal strip $P=\{w:\mathrm{\mid }\mathrm{Im}w\mathrm{\mid }<\pi \}$, with inverse $\exp {\mathrm{\mid }}_P$; thus $\mathrm{Log}$ is holomorphic, $\exp (\mathrm{Log}Z)=Z$ for $Z$ in the slit plane, and $\mathrm{Log}(\exp w)=w$ whenever $\mathrm{\mid }\mathrm{Im}w\mathrm{\mid }<\pi$ [F6].

The preimages of the two bases. For $w\in \mathbb{C}$ one has $\mathrm{\mid }\exp w\mathrm{\mid }=e^{\mathrm{Re}w}$ [F2], so $\exp w\in A_r$ if and only if $r<e^{\mathrm{Re}w}<1$, if and only if $\log r<\mathrm{Re}w<0$, and $\exp w\in {\mathbb{D}}^{\ast }$ if and only if $0<e^{\mathrm{Re}w}<1$, that is $\mathrm{Re}w<0$; the equivalences use that $x\mapsto e^x$ is strictly increasing with inverse $\log$ [F3, F4, F5]. Hence ${\exp }^{-1}(A_r)=S_r$ and ${\exp }^{-1}({\mathbb{D}}^{\ast })=H_{-}$.

Translation invariance. For each $k\in \mathbb{Z}$ the translation ${\tau }_k(w):=w+2\pi ik$ preserves real parts, hence maps $S_r$ onto $S_r$ and $H_{-}$ onto $H_{-}$, with inverse ${\tau }_{-k}$; and $\exp \circ {\tau }_k=\exp$ on all of $\mathbb{C}$ because $2\pi ik\in \ker \exp$ [F1]. In particular $\tau :={\tau }_1$ is the translation by $2\pi i$.

The strip is biholomorphic to the half-plane. Define $\phi (w):=\exp (i\pi (w-\log r)\mathrm{/}L)$ on $\mathbb{C}$. For $w=x+iy\in S_r$ one has $\phi (w)=e^{-\pi y\mathrm{/}L}{e}^{i\pi (x-\log r)\mathrm{/}L}$ [F2], where the modulus is positive and the argument $\pi (x-\log r)\mathrm{/}L$ lies in $(0,\pi )$, so $\phi (w)\in \mathbb{H}$. Conversely, for $Z\in \mathbb{H}$ put $\psi (Z):=\log r-\frac{iL}{\pi }\mathrm{Log}Z$; since $\mathbb{H}$ is contained in the slit plane, $\psi$ is holomorphic [F6, F18], and writing $\mathrm{Log}Z=u+iv$ with $v\in (-\pi ,\pi )$ one has $\mathrm{Im}Z=e^u\sin v>0$, so $v\in (0,\pi )$ and $\psi (Z)=\log r+\frac{Lv}{\pi }-\frac{iLu}{\pi }$ has real part in $(\log r,0)$, that is $\psi (Z)\in S_r$. The identity $\frac{i\pi (\psi (Z)-\log r)}{L}=\mathrm{Log}Z$ gives $\phi (\psi (Z))=\exp (\mathrm{Log}Z)=Z$ [F6], and for $w\in S_r$ the number $\frac{i\pi (w-\log r)}{L}$ lies in $P$ and exponentiates to $\phi (w)$, so it equals $\mathrm{Log}\phi (w)$ by the injectivity of $\exp {\mathrm{\mid }}_P$ in [F6], and therefore $\psi (\phi (w))=w$. Thus $\phi :S_r\to \mathbb{H}$ is a bijection, holomorphic with holomorphic inverse: a biholomorphism [F18].

Both restrictions are onto. Let $z\in A_r$; by [F17] there is $\theta \in [0,2\pi )$ with $z=\mathrm{\mid }z\mathrm{\mid }(\cos \theta +i\sin \theta )$, and by [F3, F5] there is a unique real $x\in (\log r,0)$ with $e^x=\mathrm{\mid }z\mathrm{\mid }$; then $\exp (x+i\theta )=e^x(\cos \theta +i\sin \theta )=z$ [F2], so $z\in \exp (S_r)$, and with step 2.1 the image of $S_r$ is exactly $A_r$. The same argument with $x<0$ and arbitrary $\theta$ gives $\exp (H_{-})={\mathbb{D}}^{\ast }$.

Biholomorphisms onto the disc. The map $w\mapsto -iw$ is a biholomorphism $H_{-}\to \mathbb{H}$ with inverse $w\mapsto iw$, since it is complex linear, bijective, and $\mathrm{Im}(-iw)=-\mathrm{Re}w>0$ exactly when $w\in H_{-}$ [F18]; and by [F14] the Cayley map $C$ is a biholomorphism $\mathbb{H}\to \mathbb{D}$. Hence $C\circ (-i\cdot )$ is a biholomorphism $H_{-}\to \mathbb{D}$ and $C\circ \phi$ is a biholomorphism $S_r\to \mathbb{D}$, compositions of biholomorphisms being biholomorphic [F18].

Evenly covered neighbourhoods over the annulus. Let $z_0\in A_r$ and, by step 3.1, choose $w_0\in S_r$ with $\exp w_0=z_0$. Since $S_r$ is open and $w_0\in S_r$, choose $0<\delta <\pi$ with $D(w_0,\delta )\subseteq S_r$, and put $U:=\exp (D(w_0,\delta ))$. Then $U$ is open, because $\exp$ is a nonconstant holomorphic function on the domain $D(w_0,\delta )$ [F7], and $z_0\in U\subseteq A_r$ by step 2.1. By [F1], ${\exp }^{-1}(U)={\bigcup }_{k\in \mathbb{Z}}(D(w_0,\delta )+2\pi ik)$: indeed $\exp w\in U$ holds exactly when $\exp w=\exp w^{\mathrm{\prime }}$ for some $w^{\mathrm{\prime }}\in D(w_0,\delta )$, that is, when $w-w^{\mathrm{\prime }}\in 2\pi i\mathbb{Z}$ for such a $w^{\mathrm{\prime }}$. By step 2.2 every sheet $D(w_0,\delta )+2\pi ik$ is contained in $S_r$, and two distinct ones are disjoint, since an element of their intersection would exhibit $w_1,w_2\in D(w_0,\delta )$ with $w_1-w_2=2\pi i(l-k)\mathrm{\ne }0$ of modulus $\ge 2\pi$, while $\mathrm{\mid }{w}_1-w_2\mathrm{\mid }<2\delta <2\pi$. Finally, $\exp$ is injective on each sheet by step 1.2 and maps it onto $U$, using $\exp (w^{\mathrm{\prime }}+2\pi ik)=\exp w^{\mathrm{\prime }}$ [F1]; hence $U$ is an evenly covered neighbourhood of $z_0$ with sheets $D(w_0,\delta )+2\pi ik$.

Simple connectivity of the covering domains. The disc $\mathbb{D}$ is simply connected [F13], hence nonempty, path connected and with trivial fundamental group [F20]; a biholomorphism is a homeomorphism, and a homeomorphism induces an isomorphism of fundamental groups [F19], so the biholomorphic images $S_r$ and $H_{-}$ are nonempty, path connected and have trivial fundamental group: they are simply connected [F19, F20, step 3.2].

Evenly covered neighbourhoods over the punctured disc. Let $z_0\in {\mathbb{D}}^{\ast }$ and choose $w_0\in H_{-}$ with $\exp w_0=z_0$ (step 3.1), and $0<\delta <\min (\pi ,-\mathrm{Re}{w}_0)$, so that $D(w_0,\delta )\subseteq H_{-}$; the same computation as in step 4.1 shows that $U:=\exp (D(w_0,\delta ))$ is an open neighbourhood of $z_0$ contained in ${\mathbb{D}}^{\ast }$ whose preimage is the disjoint union of the sheets $D(w_0,\delta )+2\pi ik$, each mapped homeomorphically onto $U$ by $\exp$.

The exponential over the annulus is a covering. The map $\exp :S_r\to A_r$ is continuous, its image is all of $A_r$ (step 3.1), and every $z_0\in A_r$ has the evenly covered neighbourhood produced in step 4.1; hence it is a covering map [F8], and ${\exp }^{-1}(A_r)=S_r$ is step 2.1.

The exponential over the punctured disc is a covering. The same argument with step 5.1 shows that $\exp :H_{-}\to {\mathbb{D}}^{\ast }$ is a covering map, and ${\exp }^{-1}({\mathbb{D}}^{\ast })=H_{-}$ is step 2.1.

The deck group over the annulus. Since $\exp :S_r\to A_r$ is a covering map (step 5.2), its deck transformations are the homeomorphisms $h:S_r\to S_r$ with $\exp \circ h=\exp$ [F9]; let $h$ be one of them. For $w\in S_r$ the equality $\exp (h(w))=\exp (w)$ gives $h(w)-w\in 2\pi i\mathbb{Z}$ [F1], and $w\mapsto h(w)-w$ is a continuous map from the connected strip $S_r$ into the discrete set $2\pi i\mathbb{Z}$, hence is constant: $h(w)=w+2\pi ik$ for a fixed $k\in \mathbb{Z}$. Conversely every ${\tau }_k$ is a homeomorphism of $S_r$ onto itself with $\exp \circ {\tau }_k=\exp$ (step 2.2). Therefore $\mathrm{Deck}(\exp {\mathrm{\mid }}_{S_r})=\{{\tau }_k:k\in \mathbb{Z}\}$, an infinite cyclic group generated by ${\tau }_1$, since ${\tau }_k\circ {\tau }_l={\tau }_{k+l}$ and ${\tau }_1^k={\tau }_k$.

The deck group over the punctured disc. Since $\exp :H_{-}\to {\mathbb{D}}^{\ast }$ is a covering map (step 6.1), the identical argument on the connected half-plane $H_{-}$ gives $\mathrm{Deck}(\exp {\mathrm{\mid }}_{H_{-}})=\{{\tau }_k:k\in \mathbb{Z}\}$, again infinite cyclic and generated by ${\tau }_1$.

Universal covers and hyperbolic type. By steps 5.2, 6.1 and 4.2 the two exponential maps are covering maps with simply connected total space, hence universal covering spaces [F10]; the complex structures of $S_r$ and $H_{-}$ as open subsets of $\mathbb{C}$ make $\exp$ holomorphic, so by the uniqueness of the lifted structure [F21] these are the holomorphic universal covers of $A_r$ and ${\mathbb{D}}^{\ast }$. The type definition [F11] then assigns to each of the two connected Riemann surfaces its unique type; since the exhibited covers are biholomorphic to $\mathbb{D}$ (steps 2.3, 3.2) and no two of the three models are biholomorphic [F13], both $A_r$ and ${\mathbb{D}}^{\ast }$ have hyperbolic universal-covering type. Under the Axiom of Choice [A1], the cover classification [F12] moreover exhibits each of the two surfaces as a quotient of $\mathbb{D}$ by a group of holomorphic automorphisms acting freely and properly discontinuously, namely the conjugates of the deck groups of steps 6.2 and 7.1 under the uniformizations of steps 2.3 and 3.2.

Conclusion. Steps 6.2 and 7.1 identify the two deck groups as the infinite cyclic groups generated by the translation $\tau (w)=w+2\pi i$, and step 7.2 identifies both base surfaces as hyperbolic; moreover, by [F22] the deck translations preserve the pulled-back Poincaré metric, its lengths and its distance of the corresponding uniformization. This proves all three assertions of the Example.