Zariski-density conjecture for singular hyperbolic metrics

The conjecture asserts that monodromy from a singular hyperbolic metric on any non-hyperbolic Riemann surface is Zariski-dense. The original work established many cases but left one subgroup-exclusion step unresolved.

Known results

• The 2020 preprint proves density for compact surfaces, $\mathbb{C}$, $\mathbb{C}\setminus\{0\}$, and punctured tori.
• For arbitrary non-hyperbolic surfaces it excludes monodromy in subgroups conjugate to $H_{2}$, $H_{2}^{\prime}$, $H_{3}$, and $L_{0}$, reducing the problem to the subgroup $L$.
• The thrice-punctured sphere remains outside that argument because its maximal abelian cover is hyperbolic.

August 2026 claimed completion

A new preprint claims that the remaining parabolic-surface obstruction has been ruled out, completing the conjecture in arbitrary non-hyperbolic cases. This is a claim from an unrefereed preprint, not yet an independently verified result.

Current status (as of August 2026): the conjecture has a claimed complete proof, but its validity remains unverified; absent confirmation, the final subgroup-exclusion step is not settled.