Perfect and Acyclic Subgroups of Finitely Presentable Groups
نویسنده
چکیده
We consider acyclic groups of low dimension. To indicate our results simply, let G′ be the nontrivial perfect commutator subgroup of a finitely presentable group G. Then def(G) ≤ 1. When def(G) = 1, G′ is acyclic provided that it has no integral homology in dimensions above 2 (a sufficient condition for this is that G′ be finitely generated); moreover, G/G′ is then Z or Z. Natural examples are the groups of knots and links with Alexander polynomial 1. We give a further construction based on knots in S × S. In these geometric examples, G′ cannot be finitely generated; in general, it cannot be finitely presentable. When G is a 3-manifold group it fails to be acyclic; on the other hand, if G′ is finitely generated it has finite index in the group of a Q-homology 3-sphere.
منابع مشابه
Subgroup Separability in Residually Free Groups
We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type FP∞ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.
متن کاملThe Schur Multiplier, Profinite Completions and Decidability
We fix a finitely presented group Q and consider short exact sequences 1 → N → Γ → Q → 1 with Γ finitely generated. The inclusion N ↪→ Γ induces a morphism of profinite completions N̂ → Γ̂. We prove that this is an isomorphism for all N and Γ if and only if Q is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residua...
متن کاملOn the Finite Presentation of Subdirect Products and the Nature of Residually Free Groups
We establish virtual surjection to pairs (VSP) as a general criterion for the finite presentability of subdirect products of groups: if Γ1, . . . ,Γn are finitely presented and S < Γ1×· · ·×Γn projects to a subgroup of finite index in each Γi × Γj , then S is finitely presentable, indeed there is an algorithm that will construct a finite presentation for S. We use the VSP criterion to character...
متن کاملDistortion of Wreath Products in Some Finitely Presented Groups
Wreath products such as Z ≀ Z are not finitely-presentable yet can occur as subgroups of finitely presented groups. Here we compute the distortion of Z ≀ Z as a subgroup of Thompson’s group F and as a subgroup of Baumslag’s metabelian group G. We find that Z ≀ Z is undistorted in F but is at least exponentially distorted in G.
متن کاملGroups Not Presentable by Products
In this paper we study obstructions to presentability by products for finitely generated groups. Along the way we develop both the concept of acentral subgroups, and the relations between presentability by products on the one hand, and certain geometric and measure or orbit equivalence invariants of groups on the other. This leads to many new examples of groups not presentable by products, incl...
متن کامل