A geometric invariant for nilpotent-by-abelian-by-finite groups (original) (raw)

On Torsion-Free Abelian k-Groups

Proceedings of the American Mathematical Society, 1987

It is shown that a knice subgroup with cardinality Ni, of a torsion-free completely decomposable abelian group, is again completely decomposable. Any torsion-free abelian fc-group of cardinality H" has balanced projective dimension < n. Introduction. Recently, Hill and Megibben introduced the concept of a knice subgroup in their study of abelian fc-groups [6] and also while considering the local Warfield groups in [5]. In this paper, we introduce a modified definition of a knice subgroup of a torsion-free abelian group. This helps us to extend the results of Hill and Megibben [6] and also simplify the proofs of their main theorems. Specifically we show that a knice subgroup with cardinality < Ni, of a torsion-free completely decomposable abelian group, is again completely decomposable. This enables us to prove that any torsion-free abelian fc-group (in particular, a separable group) of cardinality < Nn has balanced projective dimension < n. All the groups that we consider here are torsion-free and abelian. We generally follow the notation and terminology of L. Fuchs [3]. Let P denote the set of all primes. By a height sequence we mean a sequence s = (sp), p G P, where each 5p is a nonnegative integer or the symbol oo. If G is a torsion-free group and x G G, then |x| denotes the height sequence of x where, for each p G P, \x\p denotes the height of x at the prime p. For any height sequence s = (sp), ps is the height sequence (tp), where tp-sp + 1 and tq = sq for all q / p. G(s) denotes the subgroup {x G G: \x\> s}. G(s*) is the subgroup generated by the set {x G G(s): J2pepi\x\p ~ sp) 's unbounded}. Two height sequences (sp) and (tp) are said to be equivalent if YlPep \sp ~ ¿p\ 18 finite.

A Group Theoretical Characterisation of S-Arithmetic Groups in Higher Rank Semi-Simple Groups

2002

We now state the main theorem proved in the present paper. There are a number of definitions which are used in the statement. They will be explained in sections 1 and 2. Main Theorem. Let Γ be a group satisfying the following conditions. (1) Γ is finitely generated and is virtually torsion free (see the end of the introduction for an explanation of this term). (2) The semi-simple dimension of Γ (denoted s.s. dim (Γ)) is finite and is positive (see sections (1.1)-(1.3) for an explanation of the terms semisimple p−dimension (denoted s.s.p-dim)) and semi-simple dimension (denoted s.s.dim). (3) The semi-simple rank of Γ (denoted s..s. rank (Γ)) is at least two : s.s. rank (Γ) ≥ 2 (see (2.1) for the definition). (4) Γ is hereditarily just infinite (i.e. if Γ ⊂ Γ is a subgroup of finite index and N ⊂ Γ is a normal subgroup of Γ , then either Γ /N is finite or N is finite). (5) Let X def = {l | l is a prime and s.s.l-dim(Γ) < s.s.dim (Γ)} (we prove, under the assumptions (1) through (4), that X is a finite set: see the proof of Theorem (1.17)). Let q be a prime with q not in X. Form the product Ω = ∈X Γ × Γ q. Let Q l (l ∈ X) be the semi-simple quotient of Γ l and Q q that of Γ q (see (1.17) for a proof that under the assumption (2), such quotients exist). Let Ω = l∈X Q l × Q q ; there is a natural map from Γ to Ω which under the assumptions (1)-(4), is a virtual inclusion. Let Γ ⊂ Λ ⊂ Ω and Λ satisfy the properties (1), (2), (3) (but not necessarily (4)) as well as the equalities s.s.p − dim(Λ) = s.s.p − dim(Γ) for every prime p. Then we assume that Λ/Γ is finite. Under these assumptions (1) through (5), Γ is virtually isomorphic to an S−arithmetic subgroup Γ of a group G, which is defined over a number field K and is absolutely simple (here, S is a finite set of places of K containing all the Archimedean ones). Moreover, S−rank (G) def = v∈S K v −rank (G) ≥ 2, Γ ⊂ G(O S) , O S = S-integers on K, and G(O S)/Γ is finite. Remark: We note (see §1 and §2) that all these assumptions (s.s. dimension being finite, semi-simple rank being at least two, and the condition (5)) are properties of a quotient of a certain prop completion of Γ, and as such, are not really dependent on a linear realisation of Γ. However, it turns out (see Theorem (1.17)) that there is a more or less canonical linear realisation of our group Γ under these assumptions. Corollary 1. Suppose Γ is a group satisfying the properties (1) through (5) of the Main Theorem. Suppose that s.s.p−dim(Γ) = s.s.dim(Γ) for every prime p (in other words, suppose that the set X of (5) of the Main Theorem is empty). Then, Γ is isomorphic to an arithmetic subgroup of a higher rank semisimple real Lie group. K such that S−rank (G) := v∈S K v −rank (G) ≥ 2. Let Γ be a subgroup of finite index in G(O s) (O S = S-integers in K). Then Γ satisfies the propertis (1) through (5). It is of interest to note that in fact, we can recover-purely from the abstract properties of Γ-the S-dimension of the semi-simple group in which Γ sits as a lattice , and also its S-rank. However, the field K over which our group G is defined, and the set S of places, are irrecoverable by our methods (our methods use the prop completion of Γ and there are non-isomorphic arithmetic groups with isomorphic profinite completions (see section (3.1)). We prove the Main Theorem in §3. In the course of the proof, we also use Theorem 2. Among the S-arithmetic groups characterised by the Main Theorem, it would be desirable to single out the non-uniform ones. In §3 we also define the notion of a u-element (following [L-M-R1]) and deduce Corollary 3. Let Γ be an abstract group possessing the properties (1) through (5) of the Main Theorem. Suppose Γ has a u−element of infinite order. Then Γ is isomorphic to an S−arithmetic non-uniform lattice.

Notes on abelian groups. II

Acta Mathematica Academiae Scientiarum Hungaricae

w 5. p.basic subgroups of arbitrary abelian groups KULIKOV [8] introduced the notion of basic subgroups of abelian p-groups which has proved to be one of the most important notions in the theory of p-groups of arbitrary power. Basic subgroups can be defined in any module over the ring of p-adic integers, or, more generally, over any discrete valuation ring. Here we want to give a generalization of basic subgroups to any group so that it coincides with the old concept whenever the group is primary. In the general case, to every prime p, one can define p-basic subgroups where in the definition the prime p plays a distinguished role. The p-basic subgroups are not isomorphic for different primes, but are uniquely determined (up to isomorphism) by the group and the prime p. We shall see that p-basic subgroups are useful in certain investigations. Let G be an arbitrary (abelian) group l and p an arbitrary, but fixed prime. We call a subset [x~]~ea of G, not containing 0, p-independent, if for any finite subset xl .... ,x~ a relation nlxl-[-... q-nkx1~ EprG

A sufficient condition for finite presentability of abelian-by-nilpotent groups

Groups, Geometry, and Dynamics, 2014

A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Mathematics Subject Classification (2010) 20F16, 20F65. in a series of papers in the late 1990s: in [5], Brookes, Roseblade and Wilson showed that a finitely presented abelian-by-polycyclic group G is necessarily nilpotent-by-nilpotent-byfinite; in then, this conclusion is sharpened to G is nilpotent, by nilpotent of class at most two, by finite.

On a new construction in group theory

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2009

This paper continues the investigation of the groups RF(G) introduced and studied in [I. M. Chiswell and T. W. Müller, A class of groups with canonical Rtree action, Springer LNM, to appear]. Two new concepts, that of a test function, and that of a pair of locally incompatible (test) functions are introduced, and their theory is developed. As application, we obtain a number of new quantitative as well as structural results concerning the groups RF(G) and their quotients RF(G)/E(G) modulo the subgroup E(G) generated by the elliptic elements. Among other things, the cardinality of RF(G) is determined, and it is shown that both RF(G) and RF(G)/E(G) contain large free subgroups, and that their abelianizations both contain a large Qvector space as direct summand.

On Torsion-by-Nilpotent Groups

Journal of Algebra, 2001

Let C C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C C are torsion-by-nilpotent, then all soluble groups of C C are torsion-by-nilpotent. From that, we deduce the following conse-Ž quence, similar to a well-known result of P. Hall 1958, Illinois J. Math. 2,. 787᎐801 : if H is a normal subgroup of a group G such that H and GrHЈ are Ž. Ž. locally finite-by-nilpotent, then G is locally finite-by-nilpotent. We give an Ž. example showing that this last statement is false when '' locally finite-by-nilpotent'' is replaced with ''torsion-by-nilpotent.''