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17
votes
Accepted
$\operatorname{Out}(F_n)$ is not linear for $n > 3$
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 …
13
votes
Accepted
Integer matrix that does not belong to a free group of rank 2
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 …
8
votes
Accepted
growth of a free group automorphism is same for finite index subgroups?
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 …
8
votes
What is a geodesic in Outer space?
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 …
3
votes
Accepted
Bases of surface groups
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 …