generally, an [**end**]{} of an infinite graph $\gh$ is an equivalence class of infinite simple paths in $\gh$, where two paths are equivalent if for every finite subgraph $K\subset \gh$, there is a connected component of $\gh\setminus K$ that intersects both paths.

Let $\gh$ be a graph. For a subgraph $H$, let its [**(internal) vertex boundary**]{} $\bdv H$ be the set of vertices of $H$ that are adjacent to some vertex not in $H$. We say that $\gh$ is [**amenable**]{} if there is an exhaustion $H_1\subset H_2\subset
\cdots\subset H_n\subset\cdots$ with $\bigcup_n H_n = \gh$ and $$\lim_{n \to\infty} {|\bdv H_n| \over |\verts(H_n)|} = 0\,.$$ Such an exhaustion (or the sequence of its vertex sets) is called a [**Følner sequence**]{}. A finitely generated group is [**amenable**]{} if its Cayley graph is amenable. For example, every finitely generated abelian group is amenable. For more on amenability of graphs and groups, see .

A [**homomorphism**]{} $\varphi: \gh_1 \to \gh_2$ from one graph $\gh_1=(\verts_1,\edges_1)$ to another $\gh_2=(\verts_2,\edges_2)$ is a pair of maps $\varphi_\verts:\verts_1\to\verts_2$ and $\varphi_\edges:\edges_1\to\edges_2$ such that $\varphi_\verts$ maps the endpoints of $e$ to the endpoints of $\varphi_\edges(e)$ for every edge $e \in \edges_1$. When both maps $\varphi_\verts:\verts_1\to\verts_2$ and $\varphi_\edges:\edges_1\to\edges_2$ are bijections, then $\varphi$ is 