  Using integral methods we recover and generalize some results by F\'{e}lix,
Halperin and Thomas on the growth of the rational homology groups of free loop
spaces, and obtain a new family of spaces whose $p$-torsion in homotopy groups
grows exponentially and satisfies Moore's Conjecture for all but finitely many
primes. In view of the results, we conjecture that there should be a strong
connection between exponential growth in the rational homotopy groups and the
$p$-torsion homotopy groups for any prime $p$.
