A Compact Cubulation of a Quadratic Free-by-Cyclic Group

Let phi be the automorphism of F(a,b,c) given by phi(a)=a, phi(b)=ab and phi(c)=bcb. We construct a compact nonpositively curved cube complex with fundamental group F(a,b,c) semidirect_phi Z. It has one vertex, seven edges, eight squares and two three-dimensional cubes. Its universal cover gives a free, proper, cocompact action on a three-dimensional CAT(0) cube complex. The construction uses an explicit four-vertex cyclic core and a cubical involution; all link and presentation calculations are supplied. This answers the literal ab formulation in frozen UnsolvedMath record AIM-GEOMETRIC_GROUP_THEORY-0024. It does not address the distinct aba formulation in the workshop report. AI-assisted, self-audited and unrefereed; no absolute-priority or proof-assistant certification claim.