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Let $\Gamma$ be a connected graph of chromatic number $d$, and let $C_n(\Gamma)$ denote the set of $n$-colorings, $n\in \mathbb{N}$. Thinking of this as a subset of $[n]^{V(\Gamma)}$ which may be endowed with the product topology, we may view $C_n(\Gamma)$ as a compact space. Suppose that $\Gamma$ has bounded valence and admits a cocompact action by $\mathrm{Aut}(\Gamma)$. For example, one could imagine that $G$ is a countable finitely-generated group and $\Gamma$ is the Cayley graph of $G$ with respect to some finite generating set, so that $G\leq \mathrm{Aut}(\Gamma)$.

Is there an $\mathrm{Aut}(\Gamma)$-invariant probability measure on $C_d(\Gamma)$ with respect to the Borel $\sigma$-algebra?

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    $\begingroup$ What's the $\sigma$-algebra on $C_n(\Gamma)$ and why is cocompactness relevant? If it's Borel/powerset don't think I see any obvious obstruction to constructing such a measure. $\endgroup$
    – Sarah Brooks
    Commented Dec 4 at 7:31
  • $\begingroup$ Yes, the Borel $\sigma$-algebra. $\endgroup$
    – Ian Agol
    Commented Dec 4 at 11:20
  • $\begingroup$ @SarahBrooks if $\Gamma$ has bounded valence and is connected then it is countable. $\endgroup$
    – Sam Hopkins
    Commented Dec 4 at 15:26
  • $\begingroup$ If $\Gamma$ is countable the sigma algebra should contain singletons, so can you not just choose a finite orbit of size $n$ (if it exists) and assign probability $1/n$ to each coloring in the orbit? $\endgroup$
    – Sarah Brooks
    Commented Dec 4 at 19:15
  • $\begingroup$ It won’t exist in general (eg if $Aut(\Gamma)$ is simple. $\endgroup$
    – Ian Agol
    Commented Dec 5 at 7:40

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