To the problem of irreducible conjugately dense subgroups of linear groups

The existence of conjugately dense subgroups in amalgamated free products is studied. A subgroup intersecting each conjugacy class in a group is said to be conjugately dense. Any modular group and any free non-Abelian group of countable or finite rank has continuum many pairwise nonconjugate conjugately dense subgroups. The subgroups decomposes into amalgamated free products, if a field has a nontrivial discrete valuation. All proper maximal reducible subgroups of the locally finite groups are conjugate to each other. The conjugately dense subgroups constructed theoretically are free and irreducible, and the Neumann's conjecture is false for the locally finite groups over a specified field.