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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

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  • $\begingroup$ 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/… $\endgroup$ – Elbabak Mar 9 '16 at 8:46
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The theorem in question is often called the Dehn-Nielsen-Baer theorem. It says that the group of outer automorphisms of a surface group is isomorphic to the mapping class group, i.e. the group of isotopy classes of diffeomorphisms of the surface. For punctured surfaces, the mapping class group is isomorphic to the subgroup of the outer automorphism group consisting of automorphisms that preserves the conjugacy classes corresponding to the loops around the punctures.

A good modern source is Farb-Margalit's book A Primer on Mapping Class Groups.

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