Classes of Groups Generalizing a Theorem of Benjamin Baumslag (original) (raw)

Residual properties of free groups II

Bulletin of the Australian Mathematical Society, 1972

In this paper it is proved that non-abelian free groups are residually [x, y \ x =1, y n =l, x = y) if and only if min{(m, k), (n, h)} is greater than 1 , and not both of (m, k) and (n, h) are 2 (where 0 is taken as greater than any natural number). The proof makes use of a result, possibly of independent interest, concerning the existence of certain automorphisms of the free group of rank two. A useful criterion which enables one to prove that non-abelian free groups are residually G for a large number of groups G is also given.

The isomorphism problem for finitely generated fully residually free groups

Journal of Pure and Applied Algebra, 2007

We prove that the isomorphism problem for finitely generated fully residually free groups (or F-groups for short) is decidable. We also show that each freely indecomposable F-group G has a decomposition that is invariant under automorphisms of G, and obtain a structure theorem for the group of outer automorphisms Out(G).

Residually finite properties of groups / Muhammad Sufi Mohd Asri

2018

In this thesis, we shall study two stronger forms of residual finiteness, namely cyclic subgroup separability and weak potency in various generalized free products and HNN extensions. Among our results, we shall show that the generalized free products and HNN extensions where the amalgamated or associated subgroups are finite, or central, or infinite cyclic, or they are direct products of an infinite cyclic subgroup with a finite subgroup, or they are finite extensions of central subgroups, are again cyclic subgroup separable or weakly potent respectively. In order to prove our results, we shall prove a criterion each for the weak potency of generalized free products and HNN extensions, but we shall use previously established criterions for cyclic subgroup separability. Finally, we shall extend our results to tree products and fundamental groups of graphs of groups.

Residual properties of free groups

Journal of Algebra, 1993

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On Finitely Generated Subgroups of Free Products

Journal of the Australian Mathematical Society, 1971

If H is a subgroup of a group G we shall say that G is H-residually finite if for every element g in G, outside H, there is a subgroup of finite index in G, containing H and still avoiding g. (Then, according to the usual definition, G is residually finite if it is E-residually finite, where E is the identity subgroup). Definitions of other terms used below may be found in § 2 or in [6].

On the generic type of the free group

Journal of Symbolic Logic, 2011

We answer a question raised by Pillay, that is whether the infinite weight of the generic type of the free group is witnessed in FomegaF_{\omega}Fomega. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally,

Ja n 20 20 NON-∀-HOMOGENEITY IN FREE GROUPS

2020

We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not ∀-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated elementary free groups, and non-abelian limit groups form special kinds of Fräıssé classes in which embeddings must preserve ∀-formulas. We also provide interesting examples of countable non-finitely generated elementary free groups.