Equations in virtually class 2 nilpotent groups

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups H_n(𝕂) over any algebraic number field 𝕂, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group H₃(ℚ).

The article continues the series of papers on the absolute of finitely generated groups. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition probabilities is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group. We divide the absolute into the Laplacian part and degenerate part and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under tak- ing some central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelinization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group). Keywords: absolute, Laplace operator, dynamic Cayley graph, nilpotent groups, Laplacian part of absolute.

We show that if the lower central series of the fundamental group of a closed oriented 3-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion 2-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental groups of closed oriented 3-manifolds. There are many different approaches to the study of 3-manifolds. There is a flourishing combinatorial school from Dehn, Papakryiakopoulos, Haken and Waldhausen to Gordon and Luecke, which has shown that many 3-manifold questions can be reduced to questions about the fundamental group. A co-equal off-shoot of the combinatorial school folds in dynamics and complex analysis. This is Thurston's program on geometrizing certain characteristic and simple pieces of 3-manifolds by showing that each is modelled locally on one of the eight 3-dimensional geometries (of which only the hyperbolic case is not fully understood). A third perspective on 3-manifolds is through quantuum field theory. The ideas of Witten, Jones, Vassiliev and many others have inspired tremendous activity and, in time, may contribute substantially to the topological understanding of 3-manifolds. This paper takes a fourth perspective by looking at a 3-manifold through nilpotent eyes, observing only the tower of nilpotent quotients of the fundamental group, but never the group itself. This point of view has a long history in the study of link complements and it arises naturally if one studies 3- and 4- dimensional manifolds together. For example, Stallings proved that for a link in S 3 certain nilpotent quotients of the fundamental group of the link complement are invariants of the topological concordance class of the link. These quotients contain the same information as Milnor's ¯ µ-invariants which are generalized linking numbers. For precise references about this area of research and the most recent applications to 4-manifolds see (5). Turaev (11) seems to have been the first to consider nilpotent quotients of closed 3-manifold rather than link complements. Much earlier, the nilpotent fundamental groups of closed 3-manifolds were classified. Thomas (10) showed in particular that statements (1) and (2) in the following Theorem 1 are equivalent.

In general, a class-preserving automorphism of generalized free products of nilpotent groups, amalgamating a cyclic normal subgroup of order 8, need not be an inner automorphism. We prove that every class-preserving automorphism of generalized free products of nitely generated nilpotent groups, amalgamating a cyclic normal subgroup of order less than 8, is inner.

For connected nilpotent groups, 7 is the lowest dimension\nwhere there are infinitely many non-isomorphic groups, and also\nwhere some groups have no discrete cocompact subgroups. Here\none infinite family of 7-dimensional connected groups is studied,\ndiscrete cocompact subgroups H are found for some of them, and then\nthe faithful simple quotients A of $C^{*}(\\roman H)$ are identified. Such A are\nshown to be isomorphic to $C^{*}$-crossed products $C^{*}(\\roman H_\n3,\\Cal C(\\Bbb T^3))$ generated\nby some intriguing effective minimal distal flows $(\\roman H_3,\\Bbb T^\n3)$, where $\\roman H_3$ is the\ndiscrete 3-dimensional Heisenberg group.

This paper is devoted to the study of global controllability of linear systems on higher dimensional Heisenberg groups . Necessary conditions and sufficient ones are provided. They are compared to the known necessary and sufficient conditions on stated in Dath and Jouan [(2016). Controllability of linear systems on low dimensional nilpotent and solvable Lie groups. Journal of Dynamical and Control Systems, 22(2), 207–225]. The notion of decoupled systems is introduced in order to get examples and counter-examples, and more precise controllability criteria are stated for them.

The power graph of a group G is the simple graph with vertex set G and two distinct vertices are adjacent if one of them is a positive power of the other. For a finite noncyclic nilpotent group G, we study the minimum degree of Under some conditions involving the prime divisors of and the Sylow subgroups of G, we identify certain vertices associated with the generators of maximal cyclic subgroups of G such that is the degree of one of them. As an application, we obtain for some classes of nilpotent groups G.

The notion ofgenus, applied to finitely generated nilpotent groups or to nilpotent spaces of finite type, was introduced by Mislin; he and the author showed how to introduce the structure of a finite abelian group into the genus if the groupN has finite commutator subgroup. An example is given of a complete genusN0,N1,...,Ns−1, which constitute a cyclic group generated byN1, with the additional property that eachNi embeds in its successor as a normal subgroup with quotient cyclic of orderl; of course,Ns−1 embeds inN0. The construction leads to the description of a family of nilpotent spacesX0,X1,...,Xs−1, all in the same genus, no two of the same homotopy type, such that eachXi covers its successor as a cyclicl-sheeted regular covering; of course,Xs−1 coversX0. Herep is a prime,n ≧ 1, ands=pn−1(p−1)/2, whilel is semiprimitive modulepn.

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial ℤ-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial ℤ-groups, we finally obtain the structure and invariants of the group G.

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

Let [Formula: see text] be a division ring with center [Formula: see text] and multiplicative group [Formula: see text], of char [Formula: see text], and with an involution *. Let [Formula: see text] be the group of unitary units of [Formula: see text], namely [Formula: see text]. We investigate various instances where the dimension [Formula: see text], and in which every non-central normal subgroup [Formula: see text] contains a free non cyclic subgroup. Among them, we consider the cases where [Formula: see text] is either generated over its center of characteristic 0 by a torsion free nilpotent group, or [Formula: see text] is the field of fractions of a group algebra [Formula: see text] of the residually torsion free nilpotent group [Formula: see text] over the field [Formula: see text] of characteristic 0 , or the field of fractions of the enveloping algebra of a locally solvable residually nilpotent Lie [Formula: see text]-algebra [Formula: see text].

The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit–Ashurst conjecture and show that the Amit–Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.

A necessary and sufficient condition is found such that all the nonlinear irreducible representations of a metabelian group are realizable in the real field, and all such groups with cyclic commutator subgroups are determined.

Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field