Hyperbolic volume, mod 2 homology, and k-freeness

We show that if $M$ is any closed, orientable hyperbolic $3$-manifold with $\mathrm{v}\mathrm{o}\mathrm{l}M\le 3.69$, we have $\mathrm{d}\mathrm{i}\mathrm{m}{H}_1(M;{\mathbf{F}}_2)\le 7$. This may be regarded as a qualitative improvement of a result due to Culler and Shalen, because the constant $3.69$ is greater than the ordinal corresponding to ${\omega }^2$ in the well-ordered set of finite volumes of hyperbolic $3$-manifolds. We also show that if $\mathrm{v}\mathrm{o}\mathrm{l}M\le 3.77$, we have ${\rm dim}
H_1(M;{\bf F}_2)\le10$. These results are applications of a new method for obtaining lower bounds for the volume of a closed, orientable hyperbolic $3$-manifold such that ${\pi }_1(M)$ is $k$-free for a given $k\ge 4$. Among other applications we show that if ${\pi }_1(M)$ is $4$-free we have $\mathrm{v}\mathrm{o}\mathrm{l}M>3.57$ (improving the lower bound of $3.44$ given by Culler and Shalen), and that if ${\pi }_1(M)$ is $5$-free we have $\mathrm{v}\mathrm{o}\mathrm{l}M>3.77$.