Continuous Aldous–Lyons conjecture
For every locally compact second-countable group , every invariant random subgroup of is co-sofic; equivalently, is a weak-* limit of invariant random subgroups of that are supported on lattices of (in the discrete case, on finite-index subgroups).
References
Primary source
Additional references
- Counterexamples to the continuous Aldous-Lyons conjecture — arXiv — Tsachik Gelander
Progress summary
An unverified preprint claims counterexamples that would disprove the continuous conjecture in several important classes of groups.
The problem concerns the continuous analogue of the Aldous–Lyons conjecture. No independently corroborated resolution was found, but a new preprint claims a construction settling it negatively in broad settings.
October 2026 counterexamples
Tsachik Gelander's preprint Counterexamples to the continuous Aldous-Lyons conjecture claims negative examples for connected simple real rank-one Lie groups, rank-one groups over non-Archimedean local fields, and automorphism groups of regular or biregular trees. If correct, this refutes the conjecture for all these classes; the claim is unverified.
Current status (as of October 2026): A preprint claims the conjecture is false in several major classes, but the claim remains unverified and no independently corroborated resolution was found.
Solutions 0
No solutions have been posted yet.