Triangular group presentation

A triangular group presentation has relators of word length at most three. Every finitely presented group admits one. For a relator $s_1\cdots s_{\ell }$ of length $\ell \ge 4$, introduce generators $y_2,\dots ,y_{\ell -2}$ and replace it byThe defining relations say $y_j=s_1\cdots s_j$; eliminating those generators by Tietze transformations recovers the original relation. Every new relator has length three. Geometrically these operations triangulate each relator disk in its presentation complex by inserting diagonals, so they preserve the fundamental group.