Hyperbolic manifolds with a large number of systoles

In this article, for any $n\ge 4$ we construct a sequence of compact hyperbolic $n$-manifolds $\{M_i\}$ with number of systoles at least as $\mathrm{v}\mathrm{o}\mathrm{l}(M_i{)}^{1+\frac{1}{3n(n+1)}-\epsilon }$ for any $\epsilon >0$. In dimension 3, the bound is improved to $\mathrm{v}\mathrm{o}\mathrm{l}(M_i{)}^{\frac{4}{3}-\epsilon }$. These results generalize previous work of Schmutz for $n=2$, and Dória-Murillo for $n=3$ to higher dimensions.