## Factors of Processes on Groups and Graphs

**Abstract :**
Let $X$ be a set with a group $G$ acting on it. We will consider the
question of when there exists a $G$-homomorphism between two
i.i.d. processes which are indexed by $X$. In the
case where $X=G$, we will see that amenable and nonamenable groups are
characterized by very different behavior with respect to this
question. We will also consider the case where $X$ is a graph.
The proofs involve applications of interesting ideas from
percolation theory.