= 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 $\ell\ge4$, introduce generators $y_2,\ldots,y_{\ell-2}$ and replace it by
$$
s_1s_2y_2^{-1},\qquad y_{j-1}s_jy_j^{-1}\ (3\le j\le\ell-2),\qquad y_{\ell-2}s_{\ell-1}s_\ell.
$$
The 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>.
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