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For random groups in the Gromov density model at $d<3/14$, we construct wallsin the Cayley complex $X$ which give rise to a non-trivial action by isometrieson a CAT(0) cube complex. This extends results of Ollivier-Wise andMackay-Przytycki at densities $d<1/5$ and $d<5/24$, respectively. We are ableto overcome one of the main combinatorial challenges remaining from the work ofMackay-Przytycki, and we give a construction that plausibly works at anydensity $d<1/4$.

Random groups at density $d<3/14$ act non-trivially on a CAT(0) cube complex