I would like to find some examples of multivalued holomorphic function $f(z)$ satisfying the following conditions. $f(z)$ has only two branch points/singularities: $z=0,1$ on the whole complex plane $\mathbb{C}$. We fix a maximal simply connected region $$O:=\mathbb{C}-\mathbb{R}_{\geq0},$$and also fix a particular single-valued branch $f^e(z)$ on $O$ of $f(z)$. (The domain of $f^e(z)$ is $O$)
Then, for any $g\in\pi_1(\mathbb{C}-\left\{0,1\right\})\cong\mathbb{Z}*\mathbb{Z}$, for each point $z_0\neq0,1$, we do the analytic extension of $f^e(z)$ near $z=z_0$ along the path corresponding to $g$ which starts and ends at $z_0$, then we get a new single-valued branch near $z=z_0$. We call it $f^{g}(z)$. Then, we have a right $\pi_1(\mathbb{C}-\left\{0,1\right\})$-action on the set of branches $\left\{f^g(z)|g\in\pi_1(\mathbb{C}-\left\{0,1\right\})\right\}$, given by $g_1.f^g(z)=f^{g_1g}(z)$.
Question: give an example of such multivalued function $f(z)$ such that this action is faithful. (Faithful means that if the action of a group element $g$ is trivial, then $g$ is trivial, i.e. $g=e$.)
Something fails: Let's fix a single-valued branch of the log function. Define $\log(z)=\log|z|+\operatorname{arg}(z)i$, where $\operatorname{arg}(z)\in[0,2\pi)$. For any $\alpha,\beta\in\mathbb{C}^\times$, consider$$ f(z)=z^\alpha(1-z)^\beta,\ \ \ \ f^e(z)=e^{\alpha\log(z)}e^{\beta\log(1-z)}. $$Then, it is clear that the commutator subgroup of $\pi_1(\mathbb{C}-\left\{0,1\right\})$ acts trivially on the set of single-valued branches. Basically, every elementary function (functions that can be written as finitely many times of $+-\times\div$ and composition of $e^{az},\log(bz),z^\alpha$) has this problem.