On Some of the Residual Properties of Finitely Generated Nilpotent Groups
نویسندگان
چکیده
In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol’s work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpotent group is either virtually abelian or is not virtually RFRS. As a corollary, we deduce that any RFRS group cannot contain a nonabelian torsion-free nilpotent group. This result also illustrates some of the interplay between residual torsion-free nilpotence and the RFRS condition. We close with a topological approach to the residual torsion–free nilpotence of free, surface and graph groups.
منابع مشابه
On the Residual Solvability of Generalized Free Products of Finitely Generated Nilpotent Groups
In this paper we study the residual solvability of the generalized free product of finitely generated nilpotent groups. We show that these kinds of structures are often residually solvable.
متن کاملOn the Conjugacy Separability in the Class of Finite P -groups of Finitely Generated Nilpotent Groups
It is proved that for any prime p a finitely generated nilpotent group is conjugacy separable in the class of finite p-groups if and only if the tor-sion subgroup of it is a finite p-group and the quotient group by the torsion subgroup is abelian. 1. Let K be a class of groups. A group G is called residual K (or K-residual) if for each non-unit element a ∈ G there is a homomorphism ϕ of G onto ...
متن کاملA characterization of finitely generated multiplication modules
Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals o...
متن کاملSome Results on Baer's Theorem
Baer has shown that, for a group G, finiteness of G=Zi(G) implies finiteness of ɣi+1(G). In this paper we will show that the converse is true provided that G=Zi(G) is finitely generated. In particular, when G is a finite nilpotent group we show that |G=Zi(G)| divides |ɣi+1(G)|d′ i(G), where d′i(G) =(d( G /Zi(G)))i.
متن کاملNilpotent Completions of Groups, Grothendieck Pairs, and Four Problems of Baumslag
Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residual...
متن کامل