Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top dimensional simplices of $A(S)$ the map $\lambda$ is induced by a homeomorphism $\Phi$ of $S$. 


**Question: How to show that $\lambda$ is induced by a homeomorphism of $S$ in the whole of $A(S)$?**


P.S: In the context I have heard about it the speaker said it followed from "Mosher's connectivity argument." I searched about it but I didn't find any reference or answer. 



Injective simplicial maps between Arc complexes
I'm not as familiar with this area as I should be, but it seems as though the desired result follows from the main theorem of the following [paper](http://journals.tubitak.gov.tr/math/issues/mat-10-34-3/mat-34-3-5-0812-16.pdf):


Irmak, McCarthy. *Injective simplicial maps of the arc complex*, Turk J Math, 34 (2010) , 339 – 354.


Here is their Theorem:



> 
> Let $S$ be a compact, connected, orientable surface of genus $g$ with $b \geq 1$ boundary components. If $\lambda : A(S) \to A(S)$ is an injective simplicial map, then $\lambda$ is induced by a homeomorphism $H : S\to S$.
> 
> 
> 


If $\lambda$ is induced by a homeomorphism on top simplices of $A(S)$, then it is at least (I hope) injective. I see that Mosher's connectivity argument is mentioned and used in this paper as "Theorem 3.9", so presumably this argument is exactly what you're looking for.


Finally, in case you are hunting for more references: I saw all of this in the lovely monograph by McCarthy and Papadopolous called *Simplicial actions of mapping class groups*. It is available [here](http://www.mth.msu.edu/~mccarthy/publications/simplicial.actions.pdf). 


