Let $X$ be a finite graph. Its fundamental group is the free group $F_n$ on (say) $n$ generators. Let further an automorphism $\phi$ of $F_n$ be given. It is not true in general that this automorphism is induced by a homeomorphism of the graph. So my question is:

*Can we always thicken the graph up such that this automorphism is induced by a homeomorphism of the thickening?*

A motivating example is the case of the automorphism $a,b\mapsto a,ab$ in which case we can choose the thickening to be a torus with a disc cut out and the homeomorphism is given by a Dehn twist along one of the generators. More generally those automorphisms for which the thickening can be chosen as a surface with boundary are called *geometric*.

However I would also allow higher dimensional thickenings.

Actually I am not sure what I mean by thickening and how far the answer depends on that choice. My first guess would be a compact CW-complex that deformation retracts onto the graph.