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.



Thickening graphs to get honest actions
Every automorphism of $F\_n$ can be realized by an automorphism of the 3-dimensional handlebody of genus $n$, obtained by attaching $n$ 1-handles to a 3-ball.


