Monodromy representations and geodesics of singular flat metrics on $\mathbb{H}$ Let $X$ be a closed Riemann surface of genus $g\ge 2$ and $\pi:\mathbb{H}\rightarrow X$ the holomorphic universal cover. Then $X$ is biholomorphic to the quotient $\mathbb{H}/\Gamma$, where $\Gamma$ is a Fuchsian group and $\rho:\pi_1(X)\rightarrow \Gamma$ is the monodromy representation.
Given $x,y\in \mathbb{H}$ denote by $\sigma$ the hyperbolic geodesic from $x$ to $y$. For every $\gamma\in \pi_1(X)$ clearly (since $\Gamma$ is a group of isometries of $\mathbb{H}$) $\rho(\gamma)(\sigma)$ is the hyperbolic geodesic from $\rho(\gamma)(x)$ to $\rho(\gamma)(y)$.
Now let $q$ be a quadratic holomorphic differential on $X$ and consider  $\mathbb{H}$ endowed with the flat singular metric $|\pi^*q|$. I am wondering if it is still true a statement similar to the preceding one. Given $x,y\in \mathbb{H}$ denote by $\tau$ the $|\pi^*q|$-geodesic from $x$ to $y$. Is it true that $\rho(\gamma)(\tau)$ (i.e. the monodromy action $\rho(\gamma)$ applied on $\tau$) is the $|\pi^*q|$-geodesic from $\rho(\gamma)(x)$ to $\rho(\gamma)(y)$?
 A: It is a very general fact about invariant metrics. Suppose that $X$ is a connected manifold equipped with a (possibly singular) Finsler  metric, so that the distance  between two points is given by 
$$
d(x,y)=inf_c L(c)
$$
where the infimum is taken over all paths $c$ in $X$ connecting $x$ to $y$ and $L(c)$ is the length of $c: [0,1]\to X$:
$$
L(c)=\int_0^1 |c'(t)|dt.
$$
(In general, one needs to make sure that singularities of the metric are mild enough for this integral to make sense, but in your case, the set of singularities is discrete, so you do not have a problem.) Assume also that
$$
\phi: X\to X
$$ 
is an isometry of your Finsler metric, i.e. $|D\phi(v)|=|v|$ for every tangent vector $v\in T_xX$ (where the metric is defined on $T_xX$). Then
$\phi$ preserves distance between points and sends geodesics of the geodesics, where geodesics are defined as (local) length-minimizers. Proving this is a straightforward exercise. 
In your case, geodesics are actually distance minimizers (since the metric defined by a quadratic differential is $CAT(0)$, the latter essentially goes back to Teichmuller's work: he did not have the notion of a $CAT(0)$ metric, but he saw that he can use a singular version of the Gauss-Bonnet formula).  
In your case, you have $\pi: X\to M$ the universal cover of your surface and take the pull-back of the singular Riemannian metric from $M$ to $X$; this pull-back is invariant under the covering transformations. Hence, covering transformations send geodesics to geodesics.  
