On the ranks of the algebraic $K$-Theory of hyperbolic groups

Let $G$ be a word hyperbolic group. We prove that the algebraic $K$-theory groups of $\text{dbZ}[G]$, $K_n(\text{dbZ}[G])$, have finite rank for all $n\in \text{dbZ}$. For a few classes of groups, we give explicit formulas for the ranks of the algebraic $K$-theory groups of their group rings.