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Let $G$ and $G'$ be finitely generated groups and $f:G\to G'$ a homomorphism. First question: for a given $f:G\to G'$ it possible to select generating sets $S\in G, S'\in G'$ so that their would be a graph morphism between their Cayley graphs $\Gamma(G,S)\to \Gamma(G',S')$?

Second question, is it possible to make the selection of a generating set $S$ of a finitely generated group $G$ canonical in such manner so that the map $G\to \Gamma(G,S)$ would extend to a functor from the category of finitely generated groups to the category of (appropriately decorated) graphs?

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    $\begingroup$ Sure, take $S = G, S' = G'$. The functorial version of the Cayley graph is the simplicial set $EG$ which gives the universal $G$-cover of the simplicial set $BG$, and with the above choice of generators the Cayley graphs you get are the $1$-skeleton of $EG$. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 27, 2013 at 22:26
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    $\begingroup$ @Michael: you have to be careful about the definition of graph (oriented? labeled? multi-edges?), and of graph homomorphism. Typical issue: if a generator is in the kernel, then the corresponding edge should map to a self-loop. In any functorial version, the notion of generating subset is bad, and should be replaced with the notion of generating family (a set $S$ along with a map to $G$ whose image is a generating subset). $\endgroup$
    – YCor
    Commented Nov 27, 2013 at 22:51
  • $\begingroup$ Perhaps I misunderstood the question, but it seems to me that the answer to the second part is clearly 'no'. Consider an automorphism of $\mathbb{Z}^2$ with a large real eigenvalue, say. This won't preserve any finite generating set. $\endgroup$
    – HJRW
    Commented Nov 28, 2013 at 3:59
  • $\begingroup$ @HJRW: Perhaps your "no" answer applies to a restricted question in which the generating sets are required to be finite. $\endgroup$
    – Lee Mosher
    Commented Nov 28, 2013 at 4:00
  • $\begingroup$ @Lee - indeed. That's how I read the second paragraph - it's not clear. $\endgroup$
    – HJRW
    Commented Nov 28, 2013 at 4:01

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