Duality groups are groups that admit a Poincaré-duality-like relationship between their cohomology and their homology twisted by a module known as the dualising module. Many interesting groups are (virtually) duality groups, like free groups and their (outer) automorphisms, mapping class groups, and the general linear groups over the integers. Knowing duality is useful for homology computations, and the dualising module often has a nice interpretation. In this talk I will discuss work with Ric Wade where we explore the relationship between groups acting on simplicial complexes that are locally Cohen-Macaulay (meaning that their local homology is concentrated in a single degree and free) and groups having duality, and the implications this has for the dualising module.