Reflection triangles in Coxeter groups and biautomaticity

A Coxeter system ( W , S ) is called affine-free if its Coxeter diagram contains no affine subdiagram of rank ≥ 3. Let ( W , S ) be a Coxeter system of finite rank (i.e. | S | is finite). The main result is that W is affine-free if and only if W has finitely many conjugacy classes of reflection triangles. This implies that the action of W on its Coxeter cubing (defined by Niblo and Reeves [G. Niblo and L. Reeves. Coxeter groups act on CAT(0) cube complexes. J. Group Theory 6 (2003), 399–413]) is cocompact if and only if ( W , S ) is affine-free. This result was conjectured in loc. cit. As a corollary, we obtain that affine-free Coxeter groups are biautomatic.