Ehrenpreis conjecture

Leon Ehrenpreis conjectured that any two closed Riemann surfaces of genus at least $2$ have finite covers that are arbitrarily close in Teichmüller distance. Kahn and Marković announced and published a proof in 2011 using good-pants homology and their Surface Subgroup Theorem.

2011 Kahn–Marković proof

The published arXiv paper proves that for every $K>1$, suitable finite covers of any two closed hyperbolic surfaces admit a $K$-quasiconformal map, thereby proving the conjecture. No counterexample, retraction, or substantive objection was found.

2026 effective refinement

The Effective Ehrenpreis Conjecture claims constants $k,\epsilon_0>0$ such that covers within Teichmüller distance $\epsilon$ have degree at most $\epsilon^{-k}$; it also claims this polynomial scale is optimal for specified arithmetic surfaces. This strengthens the theorem quantitatively rather than changing its settled status.

Current status (as of August 2026): The closed-surface Ehrenpreis conjecture is resolved by the Kahn–Marković proof; the effective degree bounds announced in 2026 are an additional claimed refinement.