Margulis’s conjecture on boundary ergodicity

Margulis’s conjecture asks whether ergodicity of a discrete subgroup’s action on the boundary of a higher-rank symmetric space forces the subgroup to be a lattice. The question originated with Schoen–Yau and was conjectured by Margulis.

Known results

• Margulis: ergodicity on the product of two boundaries implies that the subgroup is a lattice, using Kazhdan’s property $(T)$.
• Fraczyk–Gelander: a uniform upper bound on the injectivity radius of the quotient also implies that the subgroup is a lattice.

August 2026 claimed counterexamples

On August 27, 2026, the preprint On some aspects of discrete groups acting ergodically on the boundary claimed counterexamples for $\mathrm{SO}(n,2)$ with $n > 2$, constructed by deforming smooth group actions. If correct, this refutes the conjecture in those dimensions; the claim is unverified.

Current status (as of August 2026): The conjecture remains unresolved in the verified literature, while an unrefereed preprint claims counterexamples for $\mathrm{SO}(n,2)$ with $n > 2$.