Criterion for properness in general locally compact groups

Determine an explicit and effective criterion for two subsets L and H of an arbitrary locally compact group G to satisfy the properness relation L pitchfork H, modulo the equivalence relation ~ between subsets of G.

Background

The paper introduces binary relations pitchfork and sim on the power set of a locally compact group G to recast properness of actions as an internal property of G rather than of the homogeneous space G/H. For reductive Lie groups, a complete criterion is given via Cartan projections (Theorem 3.15).

However, beyond the reductive setting, the general problem of characterizing when L pitchfork H holds (modulo sim) has not been resolved. The authors emphasize that by "criterion" they mean a practical, effective test rather than a purely formal restatement.

References

Problem~\ref{prob:pitchfork} remains open for general Lie groups.

— Proper Actions and Representation Theory  (2506.15616 - Kobayashi, 18 Jun 2025) in Problem (Properness Criterion), Section 3.4; open status noted in Section 3.6

He also conjectured that Question~\ref{question:ci} has an affirmative answer when $G$ is a 1-connected nilpotent Lie group and $L,H$ are connected closed subgroups of $G$.

— Proper Actions on Homogeneous Spaces of the Upper Triangular Matrix Group via Coarse Geometry  (2608.25785 - Nagaya, 26 Aug 2026) in Section 1, Introduction, Historical Background