An uncountable family of finitely generated residually finite groups (Q2078816)

In this paper, the authors construct uncountably many non-isomorphic finitely generated residually finite groups. These groups are amalgamated free products \(F*_HF\) of a free group \(F\) of rank 2, where \(H\) is an infinitely generated subgroup. Although these groups are not the first family of uncountably many isomorphism classes of finitely generated residually finite groups, these groups have a simple structure and satisfy nice properties, such as being torsion-free. The authors show that these groups are pairwise non-isomorphic by computing the homologies of certain quotients of the groups. Lastly, the authors use their constructed groups to give uncountably many non-isomorphic finitely generated residually finite groups with unsolvable word problem.