Torsion-free Nilpotent Research Articles

Functional equations for zeta functions of groups and rings

We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively.

Some reflections on proving groups residually torsion-free nilpotent. I

Some reflections on proving groups residually torsion-free nilpotent. I

Polynomial Automorphisms of Soluble Groups

A polynomial automorphism of a group G is an automorphism of the form . When G is polycyclic, we prove that the set PAut(G) of all polynomial automorphisms forms a polycyclic subgroup of the automorphism group Aut(G). When G is soluble, PAut(G) is not necessarily a subgroup, but we show that the subgroup generated by PAut(G) is soluble. We also give the general form of an element of PAut(G) when G is a torsion-free non-abelian metabelian nilpotent group.

On pro-p analogues of limit groups via extensions of centralizers

We begin a study of a pro-p analogue of limit groups via extensions of centralizers and call \({\mathcal{L}}\) this new class of pro-p groups. We show that the pro-p groups of \({\mathcal{L}}\) have finite cohomological dimension, type FP∞ and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion free nilpotent) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-p group in the class \({\mathcal{L}}\) is either free pro-p or abelian.

On Residual Properties of Pure Braid Groups of Closed Surfaces

We prove that pure braid groups of closed surfaces are almost-direct products of residually torsion free nilpotent groups and hence residually torsion free nilpotent. As a corollary, we prove also that braid groups on 2 strands of closed surfaces are residually nilpotent.

Partially commutative metabelian groups: centralizers and elementary equivalence

For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory.

VIRTUAL PROPERTIES OF CYCLICALLY PINCHED ONE-RELATOR GROUPS

We prove that the amalgamated product of free groups with cyclic amalgamations satisfying certain conditions are virtually free-by-cyclic. In case the cyclic amalgamated subgroups lie outside the derived group such groups are free-by-cyclic. Similarly a one-relator HNN-extension in which the conjugated elements either coincide or are independent modulo the derived group is shown to be free-by-cyclic. In general, the amalgamated product of free groups with cyclic amalgamations is free-by-(torsion-free nilpotent). The special case of the double of a free group amalgamating a cyclic subgroup is shown to be virtually free-by-abelian. Analagous results are obtained for certain one-relator HNN-extensions.

Torsion-free constructive nilpotent R p -groups

We consider a torsion-free nilpotent Rp-group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian Rp-group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.

Metabelian Product of a Free Nilpotent Group with a Free Abelian Group

If two groups are residually-𝑃, their free product is not necessarily so; however, it is known that the free product of residually torsion-free nilpotent groups is again residually torsion-free nilpotent. In this paper it is shown that the free metabelian product of a free nilpotent group of class two with a free abelian group is residually torsion-free nilpotent.

Twisted conjugacy classes in nilpotent groups

A group is said to have the $R_\infty$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_\infty$ property when $G$ is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer $n\ge 5$, there is a compact nilmanifold of dimension $n$ on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the $R_\infty$ property. The $R_{\infty}$ property for virtually abelian and for $\mathcal C$-nilpotent groups are also discussed.

On the linearity of torsion-free nilpotent groups of finite Morley rank

It is proved that every torsion-free nilpotent group of finite Morley rank is isomorphic to a matrix group over a field of characteristic zero.

Orbit-counting for nilpotent group shifts

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens’ theorem for the full G G -shift for a finitely-generated torsion-free nilpotent group G G . Using bounds for the Möbius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ ∑ | τ | ≤ N 1 e h | τ | ∼ C N α ( log ⁡ N ) β \sum _{\vert \tau \vert \le N}\frac {1}{e^{h\vert \tau \vert }}\sim CN^{\alpha }(\log N)^{\beta } \] where | τ | \vert \tau \vert is the cardinality of the finite orbit τ \tau and h h denotes the topological entropy. For the usual orbit-counting function we find upper and lower bounds, together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions.

Dominions of universal algebras and projective properties

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ? A such that every pair of homomorphisms f, g: A ? ? {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group ?H, a? generated by one element modulo H.

Invariant group orderings and Galois conjugates

This paper investigates conditions under which a given automorphism of a residually torsion-free nilpotent group respects some ordering of the group. For free groups and surface groups, this has relevance to ordering the fundamental groups of three-dimensional manifolds which fiber over the circle.

Recognizing powers in nilpotent groups and nilpotent images of free groups

AbstractAn element in a free group is a proper power if and only if it is a proper power in every nilpotent factor group. Moreover there is an algorithm to decide if an element in a finitely generated torsion-free nilpotent group is a proper power.

Geometric equivalence of groups

Based on the notion of geometric equivalence of groups, new classes of groups, namely, geometric varieties of groups, are defined. Some properties of such classes, including their relation to quasi-varieties and prevarieties of groups, are studied. Examples of torsion free nilpotent groups that are geometrically nonequivalent to their minimal completions, as well as an example of centrally metabelian groups that are geometrically nonequivalent but generate equal quasi-varieties, are given.

The isomorphism problem for residually torsion-free nilpotent groups

Both the conjugacy and isomorphism problems for finitely generated nilpotent groups are recursively solvable. In some recent work, the first author, with a tiny modification of work in the second author's thesis, proved that the conjugacy problem for finitely presented, residually torsion-free nilpotent groups is recursively unsolvable. Here we complete the algorithmic picture by proving that the isomorphism problem for such groups is also recursively unsolvable.

Decompositions of finitely generated nilpotent groups of class 2

In this paper we prove that rational indecomposability is a genus property for finitely generated torsion-free nilpotent groups of class 2. We use this result to determine the genus of finitely generated torsion-free nilpotent groups of class 2 which decompose as a direct product of rationally indecomposable groups.

Generalized Free Products of Residually $P$-Finite Groups

In this note, we characterize the residual p-finiteness of generalized free products and tree products of certain residually p-finite groups with non-trivial center amalgamating infinite cyclic subgroups and the tree products of certain one-relator groups. We then apply our results to tree products of finitely generated torsion-free nilpotent groups and free groups.

SUBSPACES, FREE SUBGROUPS AND A THEOREM OF STALLINGS

There is a simple group-theoretic formula for the second integral homology group of a group. This is an abelian group and there is an analogous formula for another abelian group, which involves a normal subgroup N of a torsion-free nilpotent group G. Properties of this abelian group translate into properties of G/N. This approach allows one to give a simple purely group-theoretic proof of an old theorem of J. R. Stallings, namely that if Γ is a group, if H1(G,ℤ) is free abelian and if H2(G,ℤ) = 0, then any subset Y of G which is independent modulo the derived group of G, freely generates a free group. The ideas used admit to considerable generalization, yielding in particular, proofs of a number of theorems of U. Stammbach.