Membership problems in nilpotent groups

Section 1 gives a brief review of known results on embeddings of solvable, nilpotent, and polycyclic groups in 2-generated groups from these classes, including the description of the author’s recently obtained solution to the Mikaelian–Ol’schanskii problem on embeddings of finitely generated solvable groups of derived length l in solvable groups of derived length l + 1 with a fixed small number of generators. Section 2 contains a somewhat more extensive review of known results on the rational subset membership problem for groups, including the presentation of the author’s recently obtained solution to the Laurie–Steinberg–Kambites–Silva–Zetsche problem of whether the membership problem is decidable for finitely generated submonoids of free nilpotent groups.

The product of subgroups membership problem for a finitely generated group 𝐺 is the decision problem, where, given two or more finitely generated subgroups K i K_{i} , i = 1 , … , m i=1,\ldots,m , of 𝐺 and a group element 𝑔, it is asked whether g ∈ ∏ i = 1 m K i g\in\prod_{i=1}^{m}K_{i} . In this paper, we prove that, for any finitely generated nilpotent group 𝑁 of class two, the product of two subgroups membership problem is decidable. Note that this problem is equivalent to the problem of determining the non-emptiness of the intersection of two cosets of finitely generated subgroups. We also show that, for a sufficiently large direct power H n \mathbb{H}^{n} of the Heisenberg group ℍ, there exists a product K = ∏ i = 1 4 K i K=\prod_{i=1}^{4}K_{i} of four finitely generated subgroups for which the membership problem is algorithmically undecidable. The proof of the last assertion is based on the undecidability of Hilbert’s 10th problem and interpretation of Diophantine equations in nilpotent groups. The question of decidability of the membership problem for a product of three finitely generated subgroups in a nilpotent group of class two remains open.

For finitely generated nilpotent groups, we employ Mal'cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations for subgroups are solved using only logarithmic space and quasilinear time. Logarithmic space presentation-uniform versions of these algorithms are provided. Compressed-word versions of the same problems, in which each input word is provided as a straight-line program, are solved in polynomial time.

An answer is given to the question of M. Lohrey and B. Steinberg on decidability of the submonoid membership problem for a finitely generated nilpotent group. Namely, a finitely generated submonoid of a free nilpotent group of class $2$ of sufficiently large rank $r$ is constructed, for which the membership problem is algorithmically undecidable. This implies the existence of a submonoid with similar property in any free nilpotent group of class $l \geqslant 2$ of rank $r$. The proof is based on the undecidability of Hilbert's tenth problem.

Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis, and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and we apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form $w=1$ but which are not groups, and examples defined by relations of the form $u=v$ where both of u and v are nonempty. As a consequence, we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form $aUb = a$ and examples of the form $aUb=aVa$ , with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form $uv^{-1}$ with $u, v$ positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids.

The submonoid and rational subset membership problems for graph groups

We establish a new worst-case upper bound on the Membership problem: We present a simple algorithm that is able to always achieve Agreement on Views within a single message latency after the network events leading to stability of the group become known to the membership servers. In contrast, all of the existing membership algorithms may require two or more rounds of message exchanges. Our algorithm demonstrates that the Membership problem can be solved simpler and more efficiently than previously believed.By itself, the algorithm may produce disagreement (that is, inconsistent, transient views) prior to the final view. Even though this is allowed by the problem specification, such views may create overhead at the application level, and are therefore undesirable.We propose a new approach for designing group membership services in which our algorithm for reaching Agreement on Views is combined with a filter-like mechanism for reducing disagreements. This approach can use the mechanisms of existing algorithms, yielding the same multi-round performance as theirs.However, the power of this approach is in being able to use other mechanisms. These can be tailored to the specifics of the deployment environments and to the desired combinations of the speed of agreement vs. the amount of preceding disagreement. We describe one mechanism that keeps the combined performance to within a single-round, and sketch another two.

Small support groups are a relatively new form of voluntary association appearing in the American landscape. Examining the larger implications of the “small group movement” for civic life has led some to see an overall positive effect on wider community activities, while others have viewed small-group participants as self-absorbed and withdrawing from participation in civic activities. Using a national sample of small-group participants, we examine the effects on civic engagement stemming from the social characteristics of participants, the type of small group (religious, secular, or mixed), and the opportunity to develop social capital through participation in small groups. Findings indicate that members of religious small groups do not actively engage in civic affairs, but members of secular and mixed groups do participate in a broad range of civic activities. Differences in social participation can be attributed to characteristics of the small groups (formalization and cohesion) and to some characteristics of members (education). When small support groups offer members the opportunity to acquire civic skills, to exchange information, and to develop self-efficacy, there is more interest and activity in civic affairs.

Notes on nilspaces: algebraic aspects, Discrete Analysis 2017:15, 59 pp. One of the fundamental insights in modern additive combinatorics is that there is a hierarchy of notions of "pseudorandomness" or "higher order Fourier uniformity" that can be applied either to subsets $A$ of an abelian group $G$, or functions $f: G \to {\bf C}$ of that abelian group. For instance, to understand the pseudorandomness of a subset $A$ of an abelian group $G$, one can count the number of parallelograms $$ (x, x+h_1, x+h_2, x+h_1+h_2)$$ that are fully contained in $A$, or (for a higher notion of pseudorandomness) instead count parallelepipeds $ (x, x+h_1, x+h_2, x+h_1+h_2, x+h_3, x+h_1+h_3,$ $x+h_2+h_3, x+h_1+h_2+h_3)$ or even higher-dimensional parallelepipeds. It turns out that the study of these parallelepipeds is of interest in its own right. For each dimension $d$, let $G^{[d]}$ denote the space of $d$-dimensional parallelepipeds in $G$; this is a certain subgroup of $G^{2^d}$. These spaces interact with each other in a number of ways: for instance, each $d-1$-dimensional face of the discrete cube $\{0,1\}^d$ induces a restriction map from $G^{[d]}$ to $G^{[d-1]}$. For instance, if $(x_1,\dots,x_8)$ lies in $G^{[3]}$, then $(x_1,x_2,x_3,x_4)$ or $(x_1,x_2,x_5,x_6)$ will lie in $G^{[2]}$. In the converse direction, we have the _corner completion_ property: if for instance one has seven vertices $(x_1,\dots,x_7)$ in $G$, with the property that all the two-dimensional subfaces such as $(x_1,x_2,x_3,x_4)$ or $(x_1,x_2,x_5,x_6)$ lie in $G^{[2]}$, then one can find an additional element $x_8$ of $G$ such that $(x_1,\dots,x_8)$ lies in $G^{[3]}$ (indeed, in this case this additional element will be unique). In papers of Host-Kra (["Parallelepipeds, Nilpotent Groups, and Gowers Norms"](https://arxiv.org/abs/math/0606004)) and Camarena-Szegedy (["Nilspaces, nilmanifolds and their morphisms"](https://arxiv.org/abs/1009.3825)), these properties of parallelepipeds were abstracted into axioms for a new type of structure, referred to as parallelepiped structures in Host-Kra and nilspaces in Camarena-Szegedy. In addition to the example given above of parallelepipeds on an abelian group $G$, another fundamental example of these structures comes from _nilmanifolds_ $G/\Gamma$, formed by quotienting a nilpotent Lie group $G$ by a lattice $\Gamma$. For instance, the analogue of parallelograms in this setting would be quadruples of the form $$ (x, g_1 x, g'_1 x, g_1 g'_1 g_2 x)$$ for $x \in G/\Gamma$, $g_1,g'_1 \in G$, and $g_2$ in the commutator subgroup $[G,G]$. Such spaces emerged naturally in an ergodic-theory context in work of Host and Kra (["Nonconventional ergodic averages and nilmanifolds"](http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p08.pdf)) and in a (nonstandard analysis) combinatorial context in work of Szegedy (["On higher order Fourier analysis"](https://arxiv.org/abs/1203.2260)), so it became natural to develop a systematic theory of such spaces, and in particular to look for a satisfactory classification of these spaces. As it turns out, the theory can be divided into two parts. The first part, which is simpler, is the purely "algebraic" theory of nilspaces, in which one does not require that the parallepiped structures are compatible in any way with a measure-theoretic or topological structure on the space. Then there is the "topological" and "measure-theoretic" part of the theory, in which one studies how the parallelepipeds interact with these structures. This division is analogous to the distinction between abstract group theory, and the theory of topological groups (and their interaction with measure-theoretic concepts such as Haar measure). In this paper, the first in a two-part series, the author systematically lays out the algebraic theory of nilspaces. The discussion follows closely the earlier work of Camarena and Szegedy, but with more detailed proofs and additional discussion of key examples. Just as nilpotent groups (or nilmanifolds) can be arranged in a hierarchy depending on the "step" or "nilpotency class" of the underlying nilpotent group, one can also assign a notion of a "step" to a nilspace, with higher step nilspaces being more complicated than lower step ones. For instance, as is shown in this paper, 1-step nilspaces are essentially the same as abelian groups. Perhaps the deepest result established in this paper is the fact that a $k$-step nilspace can always be viewed as an "abelian extension" of a $k-1$-step nilspace, in much the same way that a $k$-step nilpotent group can be viewed as a central extension of a $k-1$-step nilpotent group, furthermore, one can associate with the extension a certain "cocycle" which determines the $k$-step nilspace completely (up to isomorphism) once the underlying $k-1$-step nilspace is specified. Conversely, every such cocycle will generate such a $k$-step nilspace. This machinery will be used in [the second part of the series](http://discreteanalysisjournal.com/article/2106-notes-on-compact-nilspaces) to study compact nilspaces.

Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in LOGSPACE. Here we follow their approach and show that all these problems are complete for the uniform circuit class TC^0 - uniformly for all r-generated nilpotent groups of class at most c for fixed r and c. Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform TC^0, while all the other problems we examine are shown to be TC^0-Turing reducible to the problem of computing greatest common divisors and expressing them as linear combinations.

In this work we investigate the group version of the well known knapsack problem (KP) in class of nilpotent groups. We show that there exists the universal input for this problem. In other word it mean that there is the two step nilpotent group Gu and parametric input In, where n is natural parameter, such that for any torsion free nilpotent group G and input I for KP there exists a natural number m such that KP for group Gu on input Im is equivalent to KP for group G on input I.

We revisit the well-known group membership problem and show how it can be considered a special case of a simple problem, the set membership problem. In the set membership problem, processes maintain a set whose elements are drawn from an arbitrary universe: They can request the addition or removal of elements to/from that set, and they agree on the current value of the set. Group membership corresponds to the special case where the elements of the set happen to be processes. We exploit this new way of looking at group membership to give a simple and succinct specification of this problem and to outline a simple implementation approach based on the state machine paradigm. This treatment of group membership separates several issues that are often mixed in existing specifications and/or implementations of group membership. We believe that this separation of concerns greatly simplifies the understanding of this problem.

In this work we investigate the group version of the well known knapsack problem in the class of nilpotent groups. The main result of this paper is that the knapsack problem is undecidable for any torsion-free group of nilpotency class 2 if the rank of the derived subgroup is at least 316. Also, we extend our result to certain classes of polycyclic groups, linear groups, and nilpotent groups of nilpotency class greater than or equal to 2.

In [5], Mao and Hassibi started the study of finite groups that violate the Ingleton inequality. They found through computer search that the smallest group that does violate it is the symmetric group of order 120. We give a general condition that proves that a group does not violate the Ingleton inequality, and consequently deduce that finite nilpotent and metacyclic groups never violate the inequality. In particular, out of the groups of order up to 120, we give a proof that about 100 orders cannot provide groups which violate the Ingleton inequality.

When leadership is conceived as a group variable, as a set of functions which must be performed in the group, the possibility of a gradient of leadership is created, i.e., leadership may be distributed among many group members or focused on one or a few members ( 11). Marked differences of opinion have been expressed as to the consequences for group functioning of various distributions of leadership. This paper will (a) examine some findings in the research literature which can be co-ordinated profitably in a frame of reference of focused leadership, (b) report research findings supporting the position that focused leadership is associated with greater group cohesiveness, and (c) in a preliminary fashion formulate some of the conditions under which varying degrees of leadership distribution are related to group cohesiveness. Bales and his associates (12) in their study of role differentiation in small groups have described an aspect of group structure, status consensus, which they find related to group effectiveness and cohesiveness. Status consensus is the degree of agreement among the group members in their rankings of all group members on such task-related criteria as guiding the group toward its goal and contributing new ideas for the group. Operationally, status consensus is defined by Kendall's coefficient of concordance, W. Agreement on the relative task-status of all group members facilitates interaction, minimizes conflict and promotes group efficiency and harmony (2, 12). On both the operational level and conceptual level, the concept status consensus involves certain assumptions regarding the structure of small groups and the relationship between group structure and group cohesiveness. First, it is assumed that a full and complete ordering of the group members on such scales as status and influence is characteristic of small groups. Such an assumption is implicit in the index of status consensus, since discrepancies at all ranks carry equal weight in determining the magnitude of W. The assumption is explicit in the conceptual interpretation of status consensus as the degree to which the group is in accord on the relative task-status of all group members. Intuitively, it seems very unlikely that small discussion groups develop such a well-differentiated task-status order that each member can be distinguished reliably from every other group member. More probable is a partial ordering of the group members. ' Portions of this paper were read at the meeting of the American Psychological Associa