I'll describe a construction of aspherical 3-dimensional pseudomanifolds containing certain homotopically essential triangle subcomplexes. These subcomplexes have fundamental groups with strong fixed point properties, thus obstructing the "cubulability" of the pseudomanifold. I'll explain what "cubulable" means, and some details of the construction, which relies on Andreev's theorem characterizing polytopes in hyperbolic space, as well as results on expander graphs by Lubotzky--Philips--Sarnak and Margulis.