Cellular waists of hyperbolic spaces

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Chinese explanation / 中文解读

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Original abstract

We find lower bounds on the topological complexity of fibers of PL and generic smooth maps $p:M^d\rightarrow\mathbb R^m$, 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.

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• Official / arXiv page
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