Computational Algebra Seminar: Holt -- On Coxeter’s Families of Group Presentations

In 1939, Coxeter first studied the family of groups defined by the              
presentations:                                                                  
                                                                                
(l,m,n;q) = < r, s | r^l, s^m, (rs)^n,[r,s]^q >.                               
                                                                                
Until recently, the finiteness of only 6 of these groups remained undecided:    
                                                                                
(2, 3, 13; 4), (3, 4, 9; 2), (3, 4, 11; 2), (3, 4, 13; 2), (3, 5, 6; 2) and     
(3, 5, 7; 2).                                                                   
                                                                                
We discuss a recent computational proof, carried out largely in Magma,          
by George Havas and myself that (2, 3, 13; 4) is finite, and that               
(3, 4, 13; 2) and (3, 5, 7; 2) are infinite.