Rank rigidity of Euclidean polyhedra

A Euclidean polyhedron (a simplicial complex whose simplices are Euclidean) of nonpositive curvature (in the sense of Alexandrov) has rank ≥ 2 if every finite geodesic segment is a side of a flat rectangle. We prove that if a three-dimensional, geodesically complete, simply connected Euclidean polyhedron X of rank ≥ 2 and of nonpositive curvature admits a cocompact and properly discontinuous group of isometries, then X is either a Riemannian product or a thick Euclidean building of type [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] or [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /].