Multiple Mixing and Rank One Group Actions Suppose $G$ is a countable, Abelian group with an element of infinite order and let ${\cal X}$ be a mixing rank one action of $G$ on a probability space. Suppose further that the F\o lner sequence $\{F_n\}$ indexing the towers of ${\cal X}$ satisfies a bounded intersection property'': there is a constant $p$ such that each $\{F_n\}$ can intersect no more than $p$ disjoint translates of $\{F_n\}$. Then ${\cal X}$ is mixing of all orders. When $G={\bf Z}$, this extends the results of Kalikow and Ryzhikov to a large class of funny'' rank one transformations. We follow Ryzhikov's joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of $k$ copies of ${\cal X}$ is necessarily product measure. This method generalizes Ryzhikov's technique. Category:28D15