... Consider the braid group $B_3= $ and the nearest neighbor random walk defined by a probability $\nu$ on $\{a,a^{-1},b,b^{-1}\}$ that generates the whole group. ... The techniques developped to solve this problem apply, mutatis mutandis, to nearest neighbor random walks on other groups: (i) Artin groups of dihedral type; (ii) free products with amalgamation in which the factor groups are plain groups and the amalgamated subgroup is finite; (iii) HNN extensions in which the base group is finite. ...