The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erd\H{o}s), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building $\mathcal{B}$ of $G$ and there is an induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed product $C^*$-algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$ depends only on $\Gamma$ and is classified by its $K$-theory. This article shows how to compute the $K$-theory of $\mathcal{A}(\Gamma)$ and of the larger class of rank two Cuntz-Krieger algebras.