Polynomially many surfaces of fixed Euler characteristic in a hyperbolic 3-manifold

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $χ$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This bound is a polynomial function of the volume of $M$, with degree that depends linearly on $|χ|$.

💡 Research Summary

The authors establish a universal polynomial upper bound on the number of compact, essential, orientable, non‑isotopic surfaces embedded in a finite‑volume hyperbolic 3‑manifold M whose Euler characteristic is at least a fixed constant χ. Specifically, for any such manifold (closed or cusped) they prove that the number of properly embedded essential surfaces with Euler characteristic ≥ χ is bounded by
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