Monodromy of the hypergeometric equation

Remarks

The group $\pi_1(X, z_0)$ is free on the classes $\gamma_0$, $\gamma_1$ of simple loops around $0$ and $1$, so $\rho$ is determined by the two matrices $M_{\gamma_0}$ and $M_{\gamma_1}$. The eigenvalues of $M_{\gamma_0}$ are $1$ and $e^{2\pi i(1 - c)}$, and those of $M_{\gamma_1}$ are $1$ and $e^{2\pi i(c - a - b)}$. A loop around $\infty$, positively oriented in the coordinate $1/z$, has eigenvalues $e^{2\pi i a}$ and $e^{2\pi i b}$.

Example: for $a = b = 1$, $c = 2$, the solutions near $z_0$ are spanned by $w_1 = 1/z$ and $w_2 = \log(1 - z)/z$, as a direct check in the equation shows. Both are single-valued near $0$, so $M_{\gamma_0}$ is the identity. Around $1$, $\log(1 - z)$ changes by $\pm 2\pi i$, so in the basis $(w_1, w_2)$ the matrix $M_{\gamma_1}$ is $\begin{pmatrix} 1 & \pm 2\pi i \\ 0 & 1 \end{pmatrix}$, with the sign fixed by the orientation. Here the monodromy group is infinite cyclic.

Non-example: continuation along a path from $z_0$ to a different point $z_1$ does not give an element of the monodromy group. It maps solutions near $z_0$ to solutions near $z_1$, which is a different space.

This is a special case of the general notion of monodromy of a linear differential equation.