Definition of the Teichmuller space via topological marking of the $\pi_1$

Let $S_0$ be a compact orientable and oriented surface, of genus $g$, fixed once for all. For a fixed point $s_0\in S_0$, one considers a symplectic basis $a_1,\ldots,a_g,b_1,\ldots,b_g$ of $\pi_1(S_0,s_0)$ such that $$(*)\qquad \pi_1(S_0,s_0)=\langle a_1,\ldots,a_g,b_1,\ldots,b_g \, \big\lvert \; \prod_{i=1}^g[a_i,b_i]=1 \rangle=:\pi_1(g,n)\;.$$

In [W], A. Weil defines the notion of Teichmüller surface: it is a pair $(X,[f])$ where $X$ is a compact Riemann surface of genus $g$ and $[f]$ is the homotopy class of a preserving-orientation homeomorphism $f: S_0\rightarrow X$.

Given such a Teichmüller surface $(X,[f])$, one obtains a topological marking of the fundamental group of $X$, that is a family of isomorphisms (up to interior automorphisms) between the abstract group $\pi_1(g,n)$ (the RHS of (*)) and the $\pi_1(X,x)$'s for $x$ varying in $X$.

Then he claims that it follows from a theorem of Dehn that one can recover $[f]$ from the topological marking it induces.

Question:

(1). to which Dehn's result refers Weil?

(2). does the corresponding statement hold true for $n$-punctured Riemann surfaces?

Precise classical and recent references would be welcome.

References:

[W] = André WEIL, Modules des surfaces de Riemann. Séminaire Bourbaki, 4 (1956-1958), Exp. No. 168, http://www.numdam.org/item?id=SB_1956-1958__4__413_0

• Teichmüller attributes the equivalence between (what I call above) a Teichmüller surface and a topological marking to Mangler (Math. Zeit. 44 (1938), 541-554. See p. 536 of ems-ph.org/books/… – Elbabak Mar 9 '16 at 8:46