2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65532This is a survey of our work on Quantum Hyperbolic Invariants (QHI) of 3-manifolds. We explain how the theory of scissors congruence classes is a powerful geometric framework for QHI and for a `Volume Conjecture' to make sense.Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper2.abs.htmlGeometric TopologyQuantum Algebra57M27, 57Q15, 57R20, 20G42text