A cubic (trivalent) graph F is said to be 4-arc-transitive if its automorphism group acts transitively on the 4-arcs of r (where a 4-arc is a sequence «;0, vv...,vi of vertices of F such that t;,_j is adjacent to vt for 1 ^ I < 4, and vt-1 ^ vi+1 for 1 < i < 4). In his investigations into graphs of this sort, Biggs defined a family of groups 4(a), for m = 3 ,4 ,5 . . . , each presented in terms...

Let $W$ be a Coxeter group. We provide precise description of the conjugacy classes in $W$, spirit Matsumoto's theorem. This extends to all groups an important result on finite by M. Geck and G. Pfeiffer from 1993. In particular, we describe cyclically reduced elements thereby proving conjecture A. Cohen 1994.

A complex hyperbolic triangle group is the group of complex hyperbolic isometries generated by complex involutions fixing three complex lines in complex hyperbolic space. Such a group is called equilateral if there is an isometry of order three that cyclically permutes the three complex lines. We consider equilateral triangle groups for which the product of each pair of involutions and the prod...