Nontriviality under the homological obstruction

Determine whether the fundamental group π₁(cone[G]) is nontrivial for every Lie group G satisfying the homological obstruction and every non-principal ultrafilter ω.

Background

The paper distinguishes the SOL obstruction from Abels’s homological obstruction. Although the homological obstruction also implies exponential Dehn-function growth and hence guarantees at least one non-simply-connected asymptotic cone, the paper notes that an example with the homological obstruction can have a nontrivial abelian asymptotic-cone fundamental group. The unresolved question is whether nontriviality holds for every asymptotic cone in this setting.

References

And is this fundamental group nontrivial if $G$ satisfies the homological obstruction (see Remark \ref{rq:obstruction_homologique})?

— 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); related Remark 1, label rq:obstruction_homologique