Deck transformations preserve the hyperbolic metric

Setup. The surface $\tilde{X}$ is simply connected and $\psi$ is a biholomorphism, the covering $p$ is holomorphic and $G=\mathrm{Deck}(p)$ consists of the homeomorphisms $h$ of $\tilde{X}$ with $p\circ h=p$; the pulled-back metric $d{s}_{\tilde{X}}={\psi }^{\ast }d{s}_{\mathbb{D}}=p^{\ast }d{s}_X$ is a conformal metric on $\tilde{X}$ with positive smooth coefficient in every chart, because $p^{\ast }d{s}_X=(\psi \circ s\circ p{)}^{\ast }d{s}_{\mathbb{D}}={\psi }^{\ast }d{s}_{\mathbb{D}}$ on each sheet $s(V)$ of a connected evenly covered $V$ and the sheets cover $\tilde{X}$, and its length ${\mathrm{\ell }}_{\tilde{X}}$ and distance $d_{\tilde{X}}$ are defined by the same integral and infimum construction as ${\mathrm{\ell }}_X,d_X$.

The conjugates are disc automorphisms. Let $h\in G$. By [F2] $h$ is a homeomorphism with $p\circ h=p$, by [F4] it is biholomorphic, and the uniformization $\psi :\tilde{X}\to \mathbb{D}$ is biholomorphic [F13]; hence ${\gamma }_h:=\psi \circ h\circ {\psi }^{-1}$ is a bijective holomorphic self-map of $\mathbb{D}$ whose inverse $\psi \circ h^{-1}\circ {\psi }^{-1}$ is holomorphic, so ${\gamma }_h$ is an automorphism of $\mathbb{D}$ [F5].

Disc automorphisms preserve the disc length element. Let $\gamma$ be an automorphism of $\mathbb{D}$. By [F6] there are $a\in \mathbb{D}$ and $\theta \in \mathbb{R}$ with $\gamma (z)=e^{i\theta }(a-z)\mathrm{/}(1-\bar{a}z)$. Direct differentiation gives ${\phi }_a^{\mathrm{\prime }}(z)=(\mathrm{\mid }a{\mathrm{\mid }}^2-1)\mathrm{/}(1-\bar{a}z{)}^2$, so with $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }{\phi }_a^{\mathrm{\prime }}\mathrm{\mid }$ and $1-\mathrm{\mid }\gamma (z){\mathrm{\mid }}^2=1-\mathrm{\mid }{\phi }_a(z){\mathrm{\mid }}^2=(1-\mathrm{\mid }a{\mathrm{\mid }}^2)(1-\mathrm{\mid }z{\mathrm{\mid }}^2)\mathrm{/}\mathrm{\mid }1-\bar{a}z{\mathrm{\mid }}^2$ one computes $2\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(z)\mathrm{\mid }\mathrm{/}(1-\mathrm{\mid }\gamma (z){\mathrm{\mid }}^2)=2\mathrm{/}(1-\mathrm{\mid }z{\mathrm{\mid }}^2)$ for every $z\in \mathbb{D}$; equivalently the pullback of the disc length element [F3] satisfies ${\gamma }^{\ast }(d{s}_{\mathbb{D}})=d{s}_{\mathbb{D}}$.

The covering is a local isometry. Let $V\subseteq X$ be connected and evenly covered with inverse sheet $s$, so $p\circ s={\mathrm{id}}_V$ [F10]; pulling back the definition $d{s}_X{\mathrm{\mid }}_V=(\psi \circ s{)}^{\ast }d{s}_{\mathbb{D}}$ of [F1] by $p$ gives $p^{\ast }(d{s}_X)=(\psi \circ s\circ p{)}^{\ast }d{s}_{\mathbb{D}}={\psi }^{\ast }(d{s}_{\mathbb{D}})=d{s}_{\tilde{X}}$ on the sheet $s(V)$, and the sheets cover $\tilde{X}$, so $p^{\ast }(d{s}_X)=d{s}_{\tilde{X}}$ on all of $\tilde{X}$. Consequently, for every piecewise $C^1$ curve $c$ in $\tilde{X}$ [F11] the projected curve $p\circ c$ is a piecewise $C^1$ curve in $X$ [F14] and ${\mathrm{\ell }}_X(p\circ c)={\mathrm{\ell }}_{\tilde{X}}(c)$, because lengths are computed from the metric that pulls back to $p^{\ast }d{s}_X=d{s}_{\tilde{X}}$.

Lifts of piecewise $C^1$ curves are piecewise $C^1$. Let $c:[a,b]\to X$ be a piecewise $C^1$ curve and $z\in p^{-1}(c(a))$. By [F8] there is a unique continuous lift $\tilde{c}$ with $\tilde{c}(a)=z$. Refine the subdivision so that each closed parameter piece is mapped by $c$ into an evenly covered open set $U$ [F10]; this is possible because the finitely many compact parameter pieces can each be covered by finitely many such preimages. The image under $\tilde{c}$ of such a piece is connected and lies in $p^{-1}(U)$, hence inside a single sheet $S$ over $U$ [F10], and there $\tilde{c}=(p{\mathrm{\mid }}_S{)}^{-1}\circ c$. The sheet inverse $(p{\mathrm{\mid }}_S{)}^{-1}$ is holomorphic: in local charts $p{\mathrm{\mid }}_S$ is an injective holomorphic map between plane domains, so by [F9] it is biholomorphic onto its open image, and these local inverses agree with $(p{\mathrm{\mid }}_S{)}^{-1}$. Hence on each piece $\tilde{c}$ is a holomorphic map composed with a $C^1$ curve, so it is $C^1$ with derivatives extending continuously to the endpoints [F11], and $\tilde{c}$ is piecewise $C^1$.

The deck group is transitive on fibres. Let $u,v\in p^{-1}(x)$. The total space $\tilde{X}$ is simply connected [F13], hence path connected, so there is a path $\eta$ in $\tilde{X}$ from $u$ to $v$; then $\alpha :=p\circ \eta$ is a loop at $x$ whose lift starting at $u$ is $\eta$ by uniqueness in [F8], so that lift ends at $v$. By [F12] the assignment carrying a loop class to the deck transformation moving the chosen point $u$ of the fibre to the lifted endpoint is an isomorphism onto $G$; applied to the class of $\alpha$ it gives $h\in G$ with $h(u)=v$.

Deck transformations preserve the pulled-back metric. For $h\in G$ the identity $d{s}_{\tilde{X}}={\psi }^{\ast }d{s}_{\mathbb{D}}$, the relation $\psi \circ h={\gamma }_h\circ \psi$ (step 1.2) and the functoriality of pullback give $h^{\ast }(d{s}_{\tilde{X}})=(\psi \circ h{)}^{\ast }d{s}_{\mathbb{D}}=({\gamma }_h\circ \psi {)}^{\ast }d{s}_{\mathbb{D}}={\psi }^{\ast }({\gamma }_h^{\ast }d{s}_{\mathbb{D}})={\psi }^{\ast }d{s}_{\mathbb{D}}=d{s}_{\tilde{X}}$ (step 1.3).

The pulled-back distance is the disc distance in $\psi$-coordinates. For a piecewise $C^1$ curve $c$ in $\tilde{X}$ the composition $\psi \circ c$ is piecewise $C^1$ in $\mathbb{D}$ [F11, F14], and computing in a holomorphic chart of $\tilde{X}$ in which $\psi$ has expression $F$ the chain rule turns the integral for ${\mathrm{\ell }}_{\tilde{X}}(c)$ into the integral for ${\mathrm{\ell }}_{\mathbb{D}}(\psi \circ c)$; hence ${\mathrm{\ell }}_{\tilde{X}}(c)={\mathrm{\ell }}_{\mathbb{D}}(\psi \circ c)$. Since $\psi$ is a bijection with holomorphic inverse, $c\mapsto \psi \circ c$ is a bijection between the piecewise $C^1$ curves from $z$ to $w$ in $\tilde{X}$ and the piecewise $C^1$ curves from $\psi (z)$ to $\psi (w)$ in $\mathbb{D}$, so taking infima as in [F3] gives $d_{\tilde{X}}(z,w)=d_{\mathbb{D}}(\psi (z),\psi (w))$ for all $z,w\in \tilde{X}$.

Lower bound for the quotient metric. Let $x,y\in X$, let $z\in p^{-1}(x)$ and $w\in p^{-1}(y)$, and put $m:={\inf }_{h\in G}{d}_{\tilde{X}}(z,hw)$. For a piecewise $C^1$ curve $c$ in $X$ from $x$ to $y$, its lift $\tilde{c}$ from $z$ is piecewise $C^1$ by step 1.5 and ends at some point of $p^{-1}(y)$, so by step 1.6 there is $h_0\in G$ with $\tilde{c}(b)=h_{0}w$; then ${\mathrm{\ell }}_X(c)={\mathrm{\ell }}_{\tilde{X}}(\tilde{c})\ge d_{\tilde{X}}(z,\tilde{c}(b))=d_{\tilde{X}}(z,h_{0}w)\ge m$, using step 1.4, the definition of $d_{\tilde{X}}$ as an infimum [F1] and the definition of $m$. Taking the infimum over all $c$ gives $d_X(x,y)\ge m$.

Length invariance. Let $h\in G$ and let $c$ be a piecewise $C^1$ curve in $\tilde{X}$; then $h\circ c$ is piecewise $C^1$ because $h$ is holomorphic [F14], and steps 2.2 and 1.3 give ${\mathrm{\ell }}_{\tilde{X}}(h\circ c)={\mathrm{\ell }}_{\mathbb{D}}(\psi \circ h\circ c)={\mathrm{\ell }}_{\mathbb{D}}({\gamma }_h\circ (\psi \circ c))={\mathrm{\ell }}_{\mathbb{D}}(\psi \circ c)={\mathrm{\ell }}_{\tilde{X}}(c)$, since ${\gamma }_h^{\ast }(d{s}_{\mathbb{D}})=d{s}_{\mathbb{D}}$.

Distance invariance. For $h\in G$ and $z,w\in \tilde{X}$, steps 2.2 and 1.2 and [F7] give $d_{\tilde{X}}(hz,hw)=d_{\mathbb{D}}(\psi (hz),\psi (hw))=d_{\mathbb{D}}({\gamma }_h(\psi z),{\gamma }_h(\psi w))=d_{\mathbb{D}}(\psi z,\psi w)=d_{\tilde{X}}(z,w)$.

Upper bound for the quotient metric. Keep $x,y,z,w$ and $m={\inf }_{h\in G}{d}_{\tilde{X}}(z,hw)$; by step 2.2 every $d_{\tilde{X}}(z,hw)=d_{\mathbb{D}}(\psi z,\psi (hw))$ is finite [F3], so $m<+\mathrm{\infty }$. Given $\epsilon >0$, the definition of infimum provides $h\in G$ with $d_{\tilde{X}}(z,hw)<m+\epsilon$, and the definition of $d_{\tilde{X}}$ as an infimum of lengths provides a piecewise $C^1$ curve $\tilde{c}$ in $\tilde{X}$ from $z$ to $hw$ with ${\mathrm{\ell }}_{\tilde{X}}(\tilde{c})<d_{\tilde{X}}(z,hw)+\epsilon$ [F1, F3]. Then $c:=p\circ \tilde{c}$ is a piecewise $C^1$ curve in $X$ from $x$ to $y$ with ${\mathrm{\ell }}_X(c)={\mathrm{\ell }}_{\tilde{X}}(\tilde{c})<m+2\epsilon$ by step 1.4, so $d_X(x,y)<m+2\epsilon$ for every $\epsilon >0$, whence $d_X(x,y)\le m$.

The formula is independent of the lifts. If $z^{\mathrm{\prime }}\in p^{-1}(x)$ and $w^{\mathrm{\prime }}\in p^{-1}(y)$ are further lifts, step 1.6 gives $z^{\mathrm{\prime }}=h^{\mathrm{\prime }}z$ and $w^{\mathrm{\prime }}=h^{\mathrm{\prime }\mathrm{\prime }}w$ with $h^{\mathrm{\prime }},h^{\mathrm{\prime }\mathrm{\prime }}\in G$, and step 3.2 gives $d_{\tilde{X}}(z^{\mathrm{\prime }},h{w}^{\mathrm{\prime }})=d_{\tilde{X}}(h^{\mathrm{\prime }}z,h{h}^{\mathrm{\prime }\mathrm{\prime }}w)=d_{\tilde{X}}(z,h^{\mathrm{\prime }-1}h{h}^{\mathrm{\prime }\mathrm{\prime }}w)$ for every $h\in G$; since $h\mapsto h^{\mathrm{\prime }-1}h{h}^{\mathrm{\prime }\mathrm{\prime }}$ is a bijection of $G$, the sets of distances coincide and their infima are equal.

Conclusion. Assertion 1 is steps 1.2 and 1.3; assertion 2 is step 2.1 for the metric, step 3.1 for lengths, step 3.2 for distances and step 2.2 for the identity $d_{\tilde{X}}(z,w)=d_{\mathbb{D}}(\psi (z),\psi (w))$; assertion 3 is the identity $p^{\ast }(d{s}_X)=d{s}_{\tilde{X}}$ and the length statement of step 1.4 together with the inequalities $d_X(x,y)\ge m$ and $d_X(x,y)\le m$ of steps 2.3 and 3.3, the number $m$ being well defined independently of the chosen lifts by step 4.1. Hence every deck transformation is a biholomorphic isometry of the pulled-back Poincaré metric and distance, and the quotient metric on $X=\tilde{X}\mathrm{/}G$ is precisely the surface Poincaré metric.