Semigroup Intersection Problems in the Heisenberg 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.

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 prove Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal W 2 , 2 W^{2,2} -regularity for p-harmonic functions in the Heisenberg group H 1 {\mathbb H}^1 for the range 17 − 1 2 ≤ p > 5 + 5 2 \frac {\sqrt {17}-1}{2} \leq p > \frac {5+\sqrt {5}}{2} .

We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of Duchin, Liang and Shapiro to finite extensions.

In the article we prove the equivalence of the concepts of equationally Noetherianity of a torsion-free two-step nilpotent group and the existence in it of an infinite chain of strictly decreasing centralizers and present the idea of a two-way connection between equationally Noetherian partially commutative two-step nilpotent groups and equationally Noetherian graphs with loops.

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.

Let π be a set of primes. We will call a group π-separated if it can be decomposed as a central product of a π-torsion group of bounded exponent and a π-radicable group. It is easy to see that an abelian π-separated group can in fact be decomposed as a direct product of bounded and π-radicable factors (Lemma 1.1 below). A poly-π-separated group is one which can be obtained from π-separated groups by forming a finite series of group extensions. We will show here:Theorem 1. Every poly-π-separated nilpotent group is π-separated.The need for such a result arises in model theory in the case that π is the set of all primes, in which case we refer simply to separated and poly-separated groups. In connection with the conjecture that ω-stable simple groups are algebraic, it is useful to have a structure theory for solvable ω-stable groups analogous to the theory available in the algebraic case over algebraically closed fields. In particular the structure of nilpotent ω-stable groups is of interest, for example in connection with the known result [Zi1], [Ne] that the derived subgroup of a connected solvable ω-stable group of finite Morley rank is nilpotent.As a model-theoretic application of Theorem 1 we obtain:Theorem 2. If G is an ω-stable nilpotent group then G may be decomposed as a central product B * D with B and D 0-definable subgroups, B torsion of bounded exponent, and D radicable. In particular, B and D are ω-stable. Furthermore, ω · rk(B ∩ D) ≤ rk D.Corollary. The ω-stable groups of finite Morley rank are exactly the central products B * D of ω-stable nilpotent groups of finite Morley rank with B torsion of bounded exponent, D radicable, and B ∩ D finite.In addition to the purely algebraic Theorem 1, these results depend on Macintyre's characterization [Mac] of the ω-stable abelian groups as exactly the separated ones.

On rectifiable measures in Carnot groups: Marstrand–Mattila rectifiability criterion

There are many decision problems in automata theory (including membership, emptiness, emptiness of intersection, inclusion and universality problems) that for some classes of tree automata are NP-hard. The study of their parameterized complexity allows us to find new bounds of their non-polynomial time algorithmic behaviors. We present results of such a study for classical tree automata (TA), rigid tree automata (RTA), tree automata with global equality and disequality (TAGED) and t-DAG automata.KeywordsModel CheckParameterized ComplexityLinear Temporal LogicFinite AutomatonGlobal ConstraintThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

In this paper, we study the capacity/entropy region of finite, directed, acyclic, multiple-sources, and multiple-sinks network by means of group theory and entropy vectors coming from groups. There is a one-to-one correspondence between the entropy vector of a collection of \(n\) random variables and a certain group-characterizable vector obtained from a finite group and \(n\) of its subgroups. We are looking at nilpotent group characterizable entropy vectors and show that they are all also abelian group characterizable, and hence they satisfy the Ingleton inequality. It is known that not all entropic vectors can be obtained from abelian groups, so our result implies that to get more exotic entropic vectors, one has to go at least to soluble groups or larger nilpotency classes. The result also implies that Ingleton inequality is satisfied by nilpotent groups of bounded class, depending on the order of the group.

Let [Formula: see text] be an algebraic number field. With [Formula: see text] we associate the ring of finite adeles [Formula: see text] In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group [Formula: see text] Specifically, we consider matrix-valued Hamiltonian operators [Formula: see text] where [Formula: see text] is the Vladimirov operator and [Formula: see text] is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by [Formula: see text] This formula is given in terms of the ordered time exponentials.

In the previous chapter, we dealt with solvable groups and the methods came from abelian groups. In this chapter, we deal with \(S{L_2}({\mathcal{O}_S})\) where \(\mathcal{O}\) is the ring of integers in an algebraic number field and S is any set of places containing all the archimedean ones. The methods here will be very different from those of the previous chapter.

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers $\overline{\mathbb{Q}}$ . Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over $\overline{\mathbb{Q}}$ , and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(ℂ) is an algebraic point. Then we obtain a lower bound for the distance between u and W⊗ℂ, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker’s method and Hirata’s reduction as well as a new arithmetic argument (Chudnovsky’s process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.