Let $G=(V,E)$ be an countably infinite, locally finite transitive graph. Say that $G$ is exchangeable if for every two vertices $v,w \in V$ there exists a graph homomorphism that maps $v$ to $w$ and $w$ to $v$. Certainly such a graph has a unimodular automorphism group. Is there an example of a graph $G$ that is not exchangeable but is unimodular?



Non-exchangeable unimodular graph
The standard Cayley graph $\Gamma$ of the Baumslag-Solitar group $BS(1,2) = \langle a,b \mid bab^{-1} = a^2 \rangle$ is non-exchangable and unimodular (as is any Cayley graph). See <http://en.wikipedia.org/wiki/Baumslag-Solitar_group> for a picture of $\Gamma$. To see non-exchangability, observe the following:


1. an edge is labeled by $a$ iff it is contained in three distinct $5$-cycles, and by $b$ iff it is contained in two distinct $5$-cycles. Hence $\mathrm{Aut}(\Gamma)$ preserves the labeling of the (unoriented) edges by $a$ or $b$.
2. any $5$-cycles contains two $a$-labeled components, one with two edges and one with a single edge. Given any $5$-cycle, an oriented edge in the cycle is of the form $(g, gb)$ iff it points towards the $a$-labeled component with a single edge. It follows that $\mathrm{Aut}(\Gamma)$ preserves the oriented labeling of the $b$-labeled edges.


Consequently, there is no automorphism of $\Gamma$ which exchanges the vertices $1$ and $b$.


