Higher-degree homotopy and homology for the k-SOL obstruction

Determine whether, when k is minimal such that a real triangulable Lie group G has the k-SOL obstruction, the degree-k homotopy and homology groups of cone[G] have continuum cardinality.

Background

For a real triangulable group, the paper generalizes the SOL obstruction by defining the k-SOL obstruction as the condition that zero lies in the convex hull of k+1 weights of the abelianization of the exponential radical. The authors ask whether the large fundamental-group and first-homology phenomena established for the SOL obstruction persist in degree k when k is minimal.

References

A natural generalization of the questions studied here is as follows: if $k$ is minimal such that $G$ has the $k$-SOL obstruction, is it true that the homotopy and homology groups of degree $k$ of $\cone[G]$ are ``very large'', say having continuum cardinality?

— Fundamental groups of asymptotic cones of Lie groups with the SOL obstruction  (2608.30767 - Velut, 31 Aug 2026) in Section 1, subsection “Open questions” (label sec:open_questions)