MathDB

Kaimanovich–Le Prince conjecture

For every N≥2N\ge 2 and every finitely supported probability measure μ\mu on SLN(R)\mathrm{SL}_N(\mathbb{R}) such that Γ=⟨supp⁡μ⟩\Gamma=\langle\operatorname{supp}\mu\rangle is discrete and Zariski-dense, the μ\mu-stationary probability measure ν\nu on PN−1(R)\mathbb{P}^{N-1}(\mathbb{R}) is singular with respect to the SO(N)\mathrm{SO}(N)-invariant probability measure mm on PN−1(R)\mathbb{P}^{N-1}(\mathbb{R}); that is, ν⊥m\nu\perp m.

References

Primary source

arXiv

Additional references

ergodic-theoryhyperbolic-geometryrandom-walks-on-groups
Posted · edited

Progress summary

Refreshed
Claimed progress

A new preprint claims the conjectured singularity in the discrete two-dimensional real linear case, but the full higher-dimensional conjecture remains open.

The conjecture predicts that hitting or stationary measures associated with finitely supported random walks on discrete subgroups of SLN(R)\mathrm{SL}_N(\mathbb{R}) are singular relative to the natural boundary measure. The new work addresses the discrete SL2(R)\mathrm{SL}_2(\mathbb{R}) case, not the entire formulation.

Known results

  • Guivarc’h–Le Jan, 1990: non-uniform lattices in PSL2(R)\mathrm{PSL}_2(\mathbb{R}).
  • Kosenko, 2023: nearest-neighbour walks on cocompact Fuchsian groups.
  • Randecker–Tiozzo: non-uniform hyperbolic lattices in POd,1(R)\mathrm{PO}_{d,1}(\mathbb{R}).
  • Kim–Zimmer, 2025, and independently Lee–Tiozzo–Van Limbeek: substantial results for infinite-covolume groups with property (T)(T), not lattices in general.

September 15, 2026 discrete SL2(R)\mathrm{SL}_2(\mathbb{R}) claim

Timothée Bénard’s preprint claims singularity of the stationary measure on the real projective line relative to rotationally invariant measure for discrete subgroups of SL2(R)\mathrm{SL}_2(\mathbb{R}). This is meaningful progress, but it does not establish the broader PSLd(R)\mathrm{PSL}_d(\mathbb{R}) conjecture, and the claim remains unverified here.

Current status (as of September 2026): The discrete SL2(R)\mathrm{SL}_2(\mathbb{R}) case is claimed in a new preprint, while the broader PSLd(R)\mathrm{PSL}_d(\mathbb{R}) conjecture remains open.

Sources

Solutions 0

No solutions have been posted yet.