On efficient presentations of the groups $text{PSL}(2,m)$

Abstract

dWe exhibit presentations of the Von Dyck groups $D(2, 3, m)‎, ‎ mge 3$‎, ‎in terms of two generators of order $m$ satisfying three relations‎, ‎one of which is Artin's braid relation‎. ‎By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups‎. ‎In the cases $m=3, ‎‎4, 5$‎, ‎these are respectively the double covers of the finite rotational tetrahedral‎, ‎octahedral and icosahedral groups‎. ‎When $mge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups‎. ‎The interesting cases arise for $mge 7$ when these groups act as discrete groups of isometries of the hyperbolic plane‎. ‎Imposing a suitable third relation we obtain three-relator presentations of $text{PSL}(2,m)$‎. ‎We discover two general formulas presenting these as factors of $D(2, 3, m)$‎. ‎The first one works for any odd $m$ and is essentially equivalent to the shortest known presentation of Sunday cite{Sunday}‎. ‎The second applies to the cases $mequivpm 2 (text{mod} 3)$‎, ‎$m ≢ 11(text{mod} 30)$‎, ‎and is substantively shorter‎. ‎Additionally‎, ‎by random search‎, ‎we find many efficient presentations of‎ ‎finite simple Chevalley groups PSL($2,q$) as factors of $D(2, 3, m)$ where $m$ divides the order of the group‎. ‎The only other simple group that we found in this way is the sporadic Janko group $J_2$‎.