  We establish a new mixing theorem for quasirandom groups (finite groups with
no low-dimensional unitary representations) $G$ which, informally speaking,
asserts that if $g, x$ are drawn uniformly at random from $G$, then the
quadruple $(g,x,gx,xg)$ behaves like a random tuple in $G^4$, subject to the
obvious constraint that $gx$ and $xg$ are conjugate to each other. The proof is
non-elementary, proceeding by first using an ultraproduct construction to
replace the finitary claim on quasirandom groups with an infinitary analogue
concerning a limiting group object that we call an \emph{ultra quasirandom
group}, and then using the machinery of idempotent ultrafilters to establish
the required mixing property for such groups. Some simpler recurrence theorems
(involving tuples such as $(x,gx,xg)$) are also presented, as well as some
further discussion of specific examples of ultra quasirandom groups.
