Definition of weakly modular complexes

Construct or identify a general definition of a weakly modular complex, meaning a contractible complex whose 1-skeleton is a weakly modular graph.

Background

Weakly modular graphs are included among the nonpositively curved spaces considered in the paper, but the corresponding complex-level notion is not established. The authors therefore impose uniform local finiteness, which guarantees finite dimensionality for any potential complex, rather than formulate the conjecture in the most general setting.

The unresolved issue is foundational: a suitable notion of weakly modular complex would allow the locally elliptic-versus-elliptic conjecture to be stated directly for contractible complexes with weakly modular 1-skeleta.

References

For item (\ref{d:NPC6}) we do not know a general definition of a `weakly modular complex', that is, a contractible complex with $1$-skeleton being a weakly modular graph -- see Question 9.20.

— Locally elliptic actions, torsion groups, and nonpositively curved spaces  (2110.12431 - Haettel et al., 2021) in Section 2, Subsection 2.1, Remark (C) following the conjecture