Artin–Tits groups are sometimes also known as generalized braid groups, because the Artin–Tits group B S n of the (finite) symmetric group is the braid group on n strands. Not all Artin–Tits groups have a natural representation in terms of geometric braids. However, the Artin–Tits group of the hyperoctahedral group S n ± ± (geometrically, the symmetry group of the n -dimensional hypercube ; combinatorially, the group of signed permutations of size n) does have such a representation: it is given by the subgroup of the braid group on n + 1 strands consisting of those braids for which a particular strand ends in the same position it started in, or equivalently as the braid group of n strands in an annular region. Moreover, the Artin–Tits group of the hyperoctahedral group S n ± ± can be written as a semidirect product of B S ~ ~ n with an infinite cyclic group. It follows that B S ~ ~ n may be interpreted as a certain subgroup consisting of geometric braids, and also that it is a linear group.