Solving the isomorphism problems for two families of parafree groups

Abstract

For any integers m,n with m≠0 and n>0, let Gm,n denote the group presented by 〈x,y,z|x=[zm,x][zn,y]〉; for any integers m,n>0, let Hm,n denote the group presented by 〈x,y,z|x=[xm,zn][y,z]〉. By investigating cohomology jump loci of irreducible GL(2,C)-character varieties, we show: if m,m′≠0, n,n′>0 and Gm′,n′≅Gm,n, then m=m′,n=n′; if m,m′,n,n′>0 and Hm′,n′≅Hm,n, then m′=m,n′=n.