A $K$-theory criterion for $p$-hyperbolicity

For a (nice enough) finite $CW$-complex, consider the sequence of non-negative integers whose $k$-th term is the number of $\mathbb{Z}$-summands appearing in the direct sum of the first $k$ homotopy groups. A famous dichotomy in rational homotopy theory says that either this sequence is bounded (hence eventually constant) or it grows exponentially. For example, this means that no finite $CW$-complex whose rational homotopy grows polynomially exists. Huang and Wu (arXiv 2017) introduced the definitions of $p$- and $\mathbb{Z}/p^r$-hyperbolicity in order to study the growth of the number of torsion summands at a given prime $p$. I will give an overview, focussing on a condition on $K$-theory which implies $p$-hyperbolicity, and deduce some examples of $p$-hyperbolic suspensions. This condition is based on work of Selick on Moore's conjecture for torsion-free suspensions.