We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants $TV_r$ and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold $M$, with (possibly empty) toroidal boundary, $\log |TV_r (M)|$ is bounded above by a function linear in $r$ and whose slope is a positive universal constant times the Gromov norm of $M$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of $6j$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $\log| TV_r (M)|\geqslant B \ r$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose $SO(3)$ Turaev-Viro invariants grow exponentially. These constructions are essential for the results of [DK:AMU] where the authors make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class gro 查看全文>>