Cellular Waists of Hyperbolic Spaces

Abstract We find lower bounds on the topological complexity of fibers of PL and generic smooth maps $p:M^{d}\rightarrow \mathbb{R}^{m}$ p : M d → R m , where $M^{d}$ M d is a closed hyperbolic manifold of large injectivity radius. More precisely, we show that if the injectivity radius of $M$ M is greater than $50\log ((n+1)!)$ 50 log ( ( n + 1 ) ! ) , then for each dimension $0< k< d-m$ 0 < k < d − m there is a point $z\in \mathbb{R}^{m}$ z ∈ R m such that any cell structure on the fiber $p^{-1}(z)$ p − 1 ( z ) has more than $n$ n cells of dimension $k$ k . The proof is based on the freedom theorem for ideals in group rings of hyperbolic groups proved in [AD26].