By Behrstock, Drutu and Mosher [BDM], we know that the (outer) automorphism groups $\mathrm{Aut}(F\_n)$ and $\mathrm{Out}(F\_n)$ of free group of rank $n$ are *not* relatively hyperbolic if $n \geq 3$ (Theorem 9.2 of [BDM]). If $n = 2$, then $\mathrm{Out}(F\_2)$ is isomorphic to $\mathrm{GL}(2,Z)$ so that it is virtually free. I hope to know what happens to $\mathrm{Aut}(F\_2)$.


Since the left transvection and right transvection commute, $\mathrm{Aut}(F\_2)$ contains a free abelian subgroup of rank two which implies that it is not hyperbolic. Then, is it relatively hyperbolic? Or is it not relatively hyperbolic?



Is the automorphism group of free group of rank two relatively hyperbolic?
The group $\mathrm{Aut}(F\_2)$ is not relatively hyperbolic. This is contained in (the proof of) Theorem 8.1 of Behrstock-Drutu-Mosher.


We first pass to $\mathrm{Aut}^+(F\_2)$, the preimage of $\mathrm{SL}(2, \mathbb{Z})$. By Remark 7.2 of Behrstock-Drutu-Mosher it is enough to consider this index two subgroup.


Let $S$ be a copy of the two-torus. Let $Z = \{x, y\} \subset S$ be a pair of points. We now define two *mapping class groups*. There is $\Gamma = \mathrm{MCG}(S, Z)$ where $Z$ is fixed setwise. This is the usual mapping class group of (isotopy classes of) orientation preserving homeomorphisms. The *pure* mapping classes are those that fix the elements of $Z$ pointwise. We denote this index two subgroup by $\Gamma' = \mathrm{PMCG}(S, Z)$.


The point-erasing map (where we erase $y$, say) gives a homomorphism
$$
\beta : \Gamma' \to \mathrm{MCG}(S, \{x\}) \cong \mathrm{MCG}(S) \cong \mathrm{SL}(2, \mathbb{Z})
$$
The kernel of $\beta$ is isomorphic to $\pi\_1(S - \{x\}, y) \cong F\_2$. This gives the *Birman short exact sequence*
$$
1 \to F\_2 \to \Gamma' \to \mathrm{SL}(2, \mathbb{Z}) \to 1
$$
It also (with a bit of work) gives an isomorphism between $\Gamma'$ and $\mathrm{Aut}^+(F\_2)$.


We now appeal to (the proof of) Theorem 8.1 of Behrstock-Drutu-Mosher. They show that $\Gamma'$ is thick of order one, and thus not relatively hyperbolic. The proof uses the connectedness of the curve complex (Harvey) and the fact that axes of pseudo-Anosov maps are *Morse* (Behrstock's thesis).


