The symmetric genus of sporadic groups

Abstract

Given a finite group G G , the symmetric genus of G G is defined to be the smallest integer g g such that G G acts faithfully on a closed orientable surface of genus g g . Previous to this work, the task of determining the symmetric genus for the sporadic simple groups had been completed for all but nine groups: J 3 {{\text {J}}_3} , McL \operatorname {McL} , Suz \operatorname {Suz} , O’N {\text {O’N}} , Co 2 \operatorname {Co}_2 , Fi 23 \operatorname {Fi}_{23} , Co 1 \operatorname {Co}_1 , B {\text {B}} , and M {\text {M}} . In the present paper the authors resolve the problem for six of these groups, viz. J 3 {{\text {J}}_3} , McL \operatorname {McL} , Suz \operatorname {Suz} , O’N {\text {O’N}} , Co 2 \operatorname {Co}_2 , and Co 1 \operatorname {Co}_1 . Significant progress is also reported for the group Fi 23 \operatorname {Fi}_{23} .