This may be very either very simple or very unknown, but here goes: Let $F_n$ be the free group on $n$ generators and $Out(F_n)$ its outer automorphism group.
Can embeddings $Out(F_n) \hookrightarrow Out(F_m)$ be characterized for $m > n$?
This may be very either very simple or very unknown, but here goes: Let $F_n$ be the free group on $n$ generators and $Out(F_n)$ its outer automorphism group.
Can embeddings $Out(F_n) \hookrightarrow Out(F_m)$ be characterized for $m > n$?
Check out
Bogopolski O.V. and Puga D.V., On embeddings of Out(F_n), the outer automorphism group of the free group of rank n, into Out(F_m) for m>n, Algebra and Logic, v. 41, no. 2 (2002), 69-73.
There is a bunch of negative results here:
1) Khramtsov has shown that there are no embeddings for $m = n+1$ (when $n>1$)
2) Bridson and Vogtmann have shown that any homomorphism $Out(F_n) \to Out(F_m)$ factors through $\mathbb{Z} / 2 \mathbb{Z}$ if
a) $ m < n, n>2$
b) $n > 8, n< m < 2n-2$ for $n$ odd, and $n < m< 2n$ for $n $ even. This is a slightly stronger version of a result of Potapchik--Rapinchuk.
3) I have a result showing that in fact all such homomorphisms factor through $\mathbb{Z} / 2 \mathbb{Z}$ if $n > 5$ and $ n < m < {n \choose 2}$.
There are two positive results as well, one by Aramayona--Leininger--Souto and one by Bogopol'skii--Puga and Bridson--Vogtmann; in both cases $m$ grows at least exponentially with $n$.
As far as I am aware, this is all that is known. I'd be happy to give you further details if you are interested.