Chain flaring and $L^2$-torsion of free-by-cyclic groups

We introduce a condition on the monodromy of a free-by-cyclic group, $G_{\varphi }$, called the chain flare condition, that implies that the $L^2$-torsion, ${\rho }^{(2)}(G_{\varphi })$, is non-zero. We conjecture that this condition holds whenever the monodromy is exponentially growing.