  We unite elements of category theory, K-theory, and geometric group theory,
by defining a class of groups called $k$-cube groups, which act freely and
transitively on the product of $k$ trees, for arbitrary $k$. The quotient of
this action on the product of trees defines a $k$-dimensional cube complex,
which induces a higher-rank graph. We make deductions about the K-theory of the
corresponding $k$-rank graph C*-algebras, and give explicit examples of
$k$-cube groups and their K-theory. We give explicit computations of K-theory
for an infinite family of $k$-rank graphs for $k\geq 3$, which is not a direct
consequence of the K\"unneth Theorem for tensor products.
