Homotopy exponents of polyhedral products

We study Moore’s conjecture and homotopy exponents for polyhedral products. For $(\underset{‾}{CA},\underset{‾}{A}{)}^K$ where each $A_i$ is finite and has torsion-free homology, we prove that if $(\underset{‾}{CA},\underset{‾}{A}{)}^K$ is rationally hyperbolic, then it has no homotopy exponent at any odd prime. Under the additional hypothesis $\mathrm{\Sigma }{A}_i$ is homotopy equivalent to a finite-type wedge of simply-connected spheres, we show Moore’s conjecture holds for $(\underset{‾}{CA},\underset{‾}{A}{)}^K$. We also give criteria such that, for a large family of polyhedral join products, the associated polyhedral products are rationally hyperbolic, mod-$p^r$ hyperbolic for all but finitely many primes, and have no homotopy exponent at all but finitely many primes.