Let $M$ be a complete orientable hyperbolique manifold with finite volume (3 dimensional in my case, typically a knot complement), that is essentially a quotient of $\mathbb{H}^3$.
Let $r(x)=d_M(x,x_0)$ for some $x_0\in M$, and $\iota_r=\inf\{\iota_x: r(x)\leq r\} $ where $\iota_x$ is the injectivity radius at $x$.
Do we have a lower bound of the form $\iota_r\geq f_M(r)$ ? I guess the decay is at most exponential, maybe even $C_Mr^{-\alpha}$, but even a power-exponential with power $<2$ would do it for my purpose -this is for stochastic analysis.