Some finitely presented nilpotent-by-abelian groups

We consider a subclass of the class of the nilpotent (of class 2)-by-abelian groups and classify the finitely presented groups in it.     

Non-amenable finitely presented torsion-by-cyclic groups

Abstract. – We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n ≫ 1 by a cyclic group, so it satisfies the identity [x,y] ⁿ = 1. Manuscrit reĉu le 8 février 2001. RID="*" ID="*"Both authors were supported in part by the NSF grant DMS 0072307. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 99-01-00894 and by the INTAS grant, the research of the second author was supported in part by the NSF grant DMS 9978802.     

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that and let Γ be a uniform lattice in G.

(a)

If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.

(b)

If CH fails, then Γ has 2^2ω asymptotic cones up to homeomorphism.

     

JSJ-splittings for finitely presented groups over slender groups

We generalize the JSJ-splitting of Rips and Sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2-ended groups and the class of all virtually Z⊕Z groups. The approach, called “track zipping”, is relatively elementary, and differs from the Rips-Sela approach in that it does not rely on the theory of R-trees but rather on an understanding of certain embedded 1-complexes (called patterns) in a presentation 2-complex for the ambient group. Oblatum 18-IV-1997 & 30-I-1998 / Published online: 18 September 1998     

JSJ-Decompositions of finitely presented groups and complexes of groups

A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips–Sela and Dunwoody–Sageev, and we construct a JSJ-splitting for any finitely presented group with respect to the class of all slender subgroups along which the group splits. Our approach relies on Haefliger’s theory of group actions on CAT(0) spaces. Submitted: October 2003 Revision: February 2005 Accepted: June 2005     

Subdirect products and formations of finite groups

Translated from Algebra i Logika, Vol. 29, No. 5, pp. 523–548, September–October, 1990.     

Subdirect decompositions of finitely generated commutative semigroups

All finitely generated commutative semigroups which do not have proper finite subdirect decompositions are determined. This yields subdirect decompositions of finitely generated commutative semigroups and some idea of their structure.     

Finite presentation of fibre products of metabelian groups

We show that if Γ is a finitely presented metabelian group, then the “untwisted” fibre product or pull-back P associated to any short exact sequence 1→N→Γ→Q→1 is again finitely presented. In contrast, if N and Q are abelian, then the analogous “twisted” fibre-product is not finitely presented unless Γ is polycyclic. Also a number of examples are constructed, including a non-finitely presented metabelian group P with finitely generated.     

Preservation and distortion of area in finitely presented groups

IfK=G where is a tame automorphism of the 1-relator groupG, then the combinatorial area of loops in a Cayley graph ofG is undistorted in a Cayley graph ofK. Examples of distortion of area in fibres of fibrations over the circle are given and a notion of exponent of area distortion is introduced and studied. The inclusion of a finitely generated abelian subgroup in the fundamental group of a compact 3-manifold does not distort area.Partially supported by NSF grant DMS-9200433.     

Subdirect products of finite groups with Hall π-subgroups

Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 311–314, February, 1996.