Cellular waists of hyperbolic spaces

We find lower bounds on the topological complexity of fibers of PL and generic smooth maps $p:M^d\to {\mathbb{R}}^m$, where $M^d$ is a closed hyperbolic manifold of large injectivity radius. More precisely, we show that if the injectivity radius of $M$ is greater than $50\log ((n+1)!)$, then for each dimension $0<k<d-m$ there is a point $z\in {\mathbb{R}}^m$ such that any cell structure on the fiber $p^{-1}(z)$ has more than $n$ cells of dimension $k$. The proof is based on a freedom theorem for ideals in group rings of hyperbolic groups proved in arXiv:2309.16791.