Hirsch Length Research Articles

Prime Ideals in Algebras Determined by Submonoids of Nilpotent Groups

The prime spectrum of the semigroup algebra K[S] of a submonoid S of a finitely generated nilpotent group is studied via the spectra of the monoid S and of the group algebra K[G] of the group G of fractions of S. It is shown that the classical Krull dimension of K[S] is equal to the Hirsch length of the group G provided that G is nilpotent of class two. This uses the fact that prime ideals of S are completely prime. An infinite family of prime ideals of a submonoid of a free nilpotent group of class three with two generators which are not completely prime is constructed. They lead to prime ideals of the corresponding algebra. Prime ideals of the monoid of all upper triangular n × n matrices with non-negative integer entries are described and it follows that they are completely prime and finite in number.

Length-based attacks in polycyclic groups

Abstract The Anshel–Anshel–Goldfeld (AAG) key-exchange protocol was implemented and studied with the braid groups as its underlying platform. The length-based attack, introduced by Hughes and Tannenbaum, has been used to cryptanalyze the AAG protocol in this setting. Eick and Kahrobaei suggest to use the polycyclic groups as a possible platform for the AAG protocol. In this paper, we apply several known variants of the length-based attack against the AAG protocol with the polycyclic group as the underlying platform. The experimental results show that, in these groups, the implemented variants of the length-based attack are unsuccessful in the case of polycyclic groups having high Hirsch length. This suggests that the length-based attack is insufficient to cryptanalyze the AAG protocol when implemented over this type of polycyclic groups. This implies that polycyclic groups could be a potential platform for some cryptosystems based on conjugacy search problem, such as non-commutative Diffie–Hellman, El Gamal and Cramer–Shoup key-exchange protocols. Moreover, we compare for the first time the success rates of the different variants of the length-based attack. These experiments show that, in these groups, the memory length-based attack introduced by Garber, Kaplan, Teicher, Tsaban and Vishne does better than the other variants proposed thus far in this context.

Geometric dimension of groups for the family of virtually cyclic subgroups

By studying commensurators of virtually cyclic groups, we prove that every elementary amenable group of finite Hirsch length h and cardinality ℵ n admits a finite‐dimensional classifying space with virtually cyclic stabilizers of dimension n + h + 2 . We also provide a criterion for groups that fit into an extension with torsion‐free quotient to admit a finite‐dimensional classifying space with virtually cyclic stabilizers. Finally, we exhibit examples of integral linear groups of type F whose geometric dimension for the family of virtually cyclic subgroups is finite but arbitrarily larger than the geometric dimension for proper actions. This answers a question posed by W. Luck.

Commensurators and classifying spaces with virtually cyclic stabilizers

By examining commensurators of virtually cyclic groups, we show that for each natural number n , any locally finite-by-virtually cyclic group of cardinality \aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n+3 . As a corollary, we prove that every elementary amenable group of finite Hirsch length and cardinality \aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers.

Dimension of elementary amenable groups

Abstract This paper has three parts. It is conjectured that for every elementary amenable group G and every non-zero commutative ring k, the homological dimension hd k (G) is equal to the Hirsch length h(G) whenever G has no k-torsion. In Part I this conjecture is proved for several classes, including the abelian-by-polycyclic groups. In Part II it is shown that the elementary amenable groups of homological dimension one are colimits of systems of groups of cohomological dimension one. In Part III the deep problem of calculating the cohomological dimension of elementary amenable groups is tackled with particular emphasis on the nilpotent-by-polycyclic case, where a complete answer is obtained over ℚ for countable groups.

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad-Nagata dimension of a connected Lie group $G$ equipped with a left-invariant Riemannian metric coincides with its topological dimension of $G/C$ where $C$ is a maximal compact subgroup. To prove it we will compute the Assouad-Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad-Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometric to any cocompact lattice on a connected Lie group.

Random equations in nilpotent groups

In this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group G. Let Sat ( G , k ) be the set of all satisfiable equations over G in k variables. For a free abelian group A m of rank m we show that the asymptotic density ρ ( Sat ( A m , k ) ) of the set Sat ( A m , k ) (in the whole space of all equations in k variables over G) is equal to 0 for k = 1 , and is equal to ζ ( k + m ) / ζ ( k ) for k ⩾ 2 . More generally, if G is a finitely generated nilpotent infinite group, then again the asymptotic density of the set Sat ( G , 1 ) is 0. For k ⩾ 2 we give robust estimates for the upper and lower asymptotic densities of the set Sat ( G , k ) . Namely, we prove that these densities lie in the interval from 1 t ( G ) ζ ( k + h ( G ) ) ζ ( k ) to ζ ( k + m ) ζ ( k ) , where h ( G ) is the Hirsch length of G, t ( G ) is the order of lower central torsion of G, and m is the torsion-free rank (Hirsch length) of the abelianization of G. This allows one to describe the asymptotic behavior of the set Sat ( G , k ) for different parameters k , m , h ( G ) , t ( G ) .

SOME CONSIDERATIONS ON THE NONABELIAN TENSOR SQUARE OF CRYSTALLOGRAPHIC GROUPS

The nonabelian tensor square G ⊗ G of a polycyclic group G is a polycyclic group and its structure arouses interest in many contexts. The same assertion is still true for wider classes of solvable groups. This motivated us to work on two levels in the present paper: on a hand, we investigate the growth of the Hirsch length of G ⊗ G by looking at that of G, on another hand, we study the nonabelian tensor product of pro–p–groups of finite coclass, which are a remarkable class of solvable groups without center, and then we do considerations on their Hirsch length. Among other results, restrictions on the Schur multiplier will be discussed.

Virtually soluble groups of type $\operatorname{FP}_∞$

We prove that a virtually soluble group G of type \operatorname{FP}_∞ admits a finitely dominated model for \underline{\mathrm{E}}G of dimension the Hirsch length of G . This implies in particular that the Brown conjecture is satisfied for virtually torsion-free elementary amenable groups.

The subgroup lattice index problem

Given a group G and subgroups X ⩾ Y , with Y of finite index in X , then in general it is not possible to determine the index | X : Y | simply from the lattice of subgroups of G . For example, this is the case when G has prime order. The purpose of this work is twofold. First we show that in any group, if the indices | X : Y | are determined for all cyclic subgroups X , then they are determined for all subgroups X . Second we show that if G is a group with an ascending normal series with factors locally finite or abelian, and if the Hirsch length of G is at least 3, then all indices | X : Y | are determined.

Geometries and infrasolvmanifolds in dimension 4

We show that every torsion-free virtually poly-Z group of Hirsch length 4 is the fundamental group of a closed 4-manifold with a geometry of solvable Lie type, by realizing each such group as a lattice subgroup of one of the corresponding isometry groups.

BREADTH IN POLYCYCLIC GROUPS

The breadth of a polycyclic group is the maximum of h(G) - h(CG(x)) for x ∈ G, where h(G) is the Hirsch length. We prove a number of results that bound the class of a finitely generated nilpotent group, and the Hirsch length of the derived group in a polycyclic group, in terms of the breadth. These results are analogues of well-known results in finite group theory.

On Bredon homology of elementary amenable groups

We show that for elementary amenable groups the Hirsch length is equal to the Bredon homological dimension. This also implies that countable elementary amenable groups admit a finite-dimensional model for $\underbar{EG}$ of dimension less than or equal to the Hirsch length plus one. Some remarks on groups of type $FP_\infty$ are also made.

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory

We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.

Asymptotic dimension of discrete groups

We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in (B-D2) to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their Hirsch length.

The Normal Zeta Function of the Free Class Two Nilpotent Group on Four Generators

We calculate explicitly the normal zeta function of the free group of class two on four generators, denoted by F2,4. This has Hirsch length ten.

Any virtually polycyclic group admits a NIL-affine crystallographic action

In this paper, we prove that any virtually polycyclic group admits a properly discontinuous and cocompact action on a simply connected, connected nilpotent Lie group N. This action is affine in the sense that the group acts as a subgroup of the affine group Aff( N)= N⋊Aut( N) of connection preserving automorphisms of N. As a consequence, we also obtain that any virtually polycyclic group admits a polynomial crystallographic action of degree bounded above by its Hirsch length.

Linear Bounds for the Degree of Subgroup Growth in Terms of the Hirsch Length

Let G be a residually-finite virtually soluble minimax group. We give upper and lower linear bounds for the degree of subgroup growth of G in terms of the Hirsch length of G .

Solvable Lie Algebras, Lie Groups and Polynomial Structures

In this paper, we study polynomial structures by starting on the Lie algebra level, then passing to Lie groups to finally arrive at the polycyclic-by-finite group level. To be more precise, we first show how a general solvable Lie algebra can be decomposed into a sum of two nilpotent subalgebras. Using this result, we construct, for any simply connected, connected solvable Lie group G of dim n, a simply transitive action on Rn which is polynomial and of degree ≤ n3. Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group Γ, which is of degree ≤ h(Γ)3 on almost the entire group (h (Γ) being the Hirsch length of Γ).

Strong genus of nilpotent groups of Hirsch length six

We give a characterization in terms of finitely many arithmetical invariants for two finitely generated, torsion-free, nilpotent groups of nilpotency class two and Hirsch length six to have isomorphic localizations at every finite set of primes. This result is obtained through adequate reduction of an arbitrary binary integral quadratic form. We derive some consequences and, in particular, we characterize completely those groups N in the above class of groups, for which the Mislin genus coincides with the strong genus (i.e. those groups N for which, whenever a nilpotent group M satisfies Mp ≅ Np for every prime p, then MP ≅ NP for every finite set of primes P).