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Besides the geodesic paths of the asymmetric metric $d(\cdot,\cdot)$ that are mentioned in other answers (namely paths such that $d(\gamma(s),\gamma(t)) = t-s$ if $s \le t$), there is another class of …

There is an embedding $\text{Aut}(F_{n-1}) \hookrightarrow \text{Out}(F_n)$ for any $n \ge 2$, as follows. First one embeds $\text{Aut}(F_{n-1}) \hookrightarrow \text{Aut}(F_n)$ by extending any aut …

While I cannot point to any place in the literature with this particular statement, I would say that it is an exercise in relative train track theory, which is the very nice normal form theory for $\t …

This can be proved using the modular diagram in the upper half plane, and the dual tree $T$ of the modular diagram (see below for a discussion of the picture in the link). The modular diagram has vert …

There is indeed a surface basis for $H$ containing $x_1,\ldots,x_k$. I'll give a topological proof, basically the same as the proof suggested in the comment of @HJRW. First, I'll give a topological re …