Solving the isomorphism problems for two families of parafree groups

For any integers m, n with m not equal 0 and n > 0, let G(m,n) denote the group presented by < x, y, z vertical bar x = [z(m), x][z(n), y]>; for any integers m, n > 0, let H-m,H-n denote the group presented by < x, y, z vertical bar x = [x(m), z(n)][y, z]>. By investigating cohomology jump loci of irreducible GL(2, C)-character varieties, we show: if m, m' not equal 0, n, n' > 0 and G(m',n') congruent to G(m,n), then m = m', n = n'; if m, m', n, n' > 0 and H-m',H-n' congruent to H-m,H-n, then m' = m, n' = n. (C) 2021 Elsevier Inc. All rights reserved.