Abelian subgroup separability of some one‐relator groups

Abstract

We prove that any group in the class of one-relator groups given by the presentation 〈a,b;[am,bn]=1〉, where m and n are integers greater than 1, is cyclic subgroup separable (or πc). We establish some suitable properties of these groups which enable us to prove that every finitely generated abelian subgroup of any of such groups is finitely separable

On the cyclic subgroup separability of the free product of two groups with commuting subgroups

“…Note that the conditions of Theorem 3.6 are true for the group G mn = a, b; [a m , b n ] = 1 with a suitable choice of numbers m and n. This group is investigated in [10,11], where the finite separability of all its cyclic subgroups is proved.…”

“…Also, she proves that if all cyclic subgroups of A and B are separable in the class of all finite groups, then G possesses the same property. At last, D. Tieudjo and D. I. Moldavanskii [10,11,12,13] consider in detail the special case of the given construction when A and B are cyclic. This paper generalizes the results of E. D. Loginova and D. Tieudjo mentioned above.…”

“…Recall (see [6, section 4.2]) that the free product of two groups A and B with commuting subgroups H A and K B, and D. I. Moldavanskii [10,11,12,13] consider in detail the special case of the given construction when A and B are cyclic.…”

“…with a suitable choice of numbers m and n. This group is investigated in [10,11], where the finite separability of all its cyclic subgroups is proved.…”

Finding and Combining Indicable Subgroups of Big Mapping Class Groups

“…The algorithmic properties for these groups (such as the word and conjugacy problem) have been well-studied [MS73,Hur76]. Residual and separability properties of such groups have also been extensively studied in [Log99,Tie05,TM10,TM08,Sok14]. This product provides a natural interpolation between free products and direct products and includes, for example, graph products of groups.…”