Nielsen Equivalence Research Articles

Rigidity of the Torelli subgroup in $\mathrm{Out}(F_{N})$

Let N\ge 4 . We prove that every injective homomorphism from the Torelli subgroup \textup{IA}_{N} to \textup{Out}(F_{N}) differs from the inclusion by a conjugation in \textup{Out}(F_{N}) . This applies more generally to the following subgroups of \textup{Out}(F_{N}) : every finite-index subgroup of \textup{Out}(F_{N}) (recovering a theorem of Farb and Handel); every subgroup of \textup{Out}(F_{N}) that contains a finite-index subgroup of one of the groups in the Andreadakis–Johnson filtration of \textup{Out}(F_{N}) ; every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich subgroup of \textup{Out}(F_{N}) –those are subgroups that contain sufficiently many twists in an appropriate sense.Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in \textup{Out}(F_{N}) ; it also implies that all subgroups in the Andreadakis–Johnson filtration of \textup{Out}(F_{N}) are co-Hopfian.We also prove the same rigidity statement for subgroups of \textup{Out}(F_{3}) which contain a power of every Nielsen transformation. This shows, in particular, that \textup{Out}(F_{3}) and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the N=3 case.

Dependence over subgroups of free groups

Given a finitely generated subgroup H of a free group F, we present an algorithm which computes [Formula: see text], such that the set of elements [Formula: see text], for which there exists a non-trivial H-equation having g as a solution is precisely the disjoint union of the double cosets [Formula: see text]. Moreover, we present an algorithm which, given a finitely generated subgroup [Formula: see text] and an element [Formula: see text], computes a finite set of elements from [Formula: see text] (of the minimum possible cardinality) generating, as a normal subgroup, the “ideal” [Formula: see text] of all “polynomials” [Formula: see text], such that [Formula: see text]. The algorithms, as well as the proofs, are based on the graph-theoretic techniques introduced by Stallings and on the more classical combinatorial techniques of Nielsen transformations. The key notion here is that of dependence of an element [Formula: see text] on a subgroup H. We also study the corresponding notions of dependence sequence and dependence closure of a subgroup.

Nielsen equivalence in Fuchsian groups

In this paper we give a complete classification of minimal generating systems in a very general class of Fuchsian groups G. This class includes for example any G which has at least seven non-conjugate cyclic subgroups of order greater than 2. In particular, the well known problematic cases where G has characteristic exponents equal to 2 are not excluded. We classify generating systems up to Nielsen equivalence; this notion is strongly related to Heegaard splittings of 3-manifolds. The results of this paper provide in particular the tools for a rather general extension of previous work of the authors and others, on the isotopy classification of such splittings in Seifert fibered spaces.

Rank and Nielsen equivalence in hyperbolic extensions

In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered [Formula: see text]-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of [Formula: see text].

Generators of split extensions of Abelian groups by cyclic groups

Let G \simeq M \rtimes C be an n -generator group which is a split extension of an Abelian group M by a cyclic group C . We study the Nielsen equivalence classes and T-systems of generating n -tuples of G . The subgroup M can be turned into a finitely generated faithful module over a suitable quotient R of the integral group ring of C . When C is infinite, we show that the Nielsen equivalence classes of the generating n -tuples of G correspond bijectively to the orbits of unimodular rows in M^{n -1} under the action of a subgroup of GL _{n - 1}(R) . Making no assumption on the cardinality of C , we exhibit a complete invariant of Nielsen equivalence in the case M \simeq R . As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag–Solitar groups, split metacyclic groups and lamplighter groups.

Analysis of secret sharing schemes based on Nielsen transformations

Abstract We investigate security properties of two secret-sharing protocols proposed by Fine, Moldenhauer, and Rosenberger in Sections 4 and 5 of [B. Fine, A. Moldenhauer and G. Rosenberger, Cryptographic protocols based on Nielsen transformations, J. Comput. Comm. 4 2016, 63–107] (Protocols I and II resp.). For both protocols, we consider a one missing share challenge. We show that Protocol I can be reduced to a system of polynomial equations and (for most randomly generated instances) solved by the computer algebra system Singular. Protocol II is approached using the technique of Stallings’ graphs. We show that knowledge of {m-1} shares reduces the space of possible values of a secret to a set of polynomial size.

On Andrews–Curtis conjectures for soluble groups

The Andrews–Curtis conjecture claims that every normally generating [Formula: see text]-tuple of a free group [Formula: see text] of rank [Formula: see text] can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing [Formula: see text] by an arbitrary finitely generated group yields natural generalizations whose study may help disprove the original and unsettled conjecture. We prove that every finitely generated soluble group satisfies the generalized Andrews–Curtis conjecture in the sense of Borovik, Lubotzky and Myasnikov. In contrast, we show that some soluble Baumslag–Solitar groups do not satisfy the generalized Andrews–Curtis conjecture in the sense of Burns and Macedońska.

Nielsen equivalence in Gupta-Sidki groups

For a group G G generated by k k elements, the Nielsen equivalence classes are defined as orbits of the action of Aut F k \textrm {Aut} F_k , the automorphism group of the free group of rank k k , on the set of generating k k -tuples of G G . Let p ≥ 3 p\geq 3 be prime and G p G_p the Gupta-Sidki p p -group. We prove that there are infinitely many Nielsen equivalence classes on generating pairs of G p G_p .

Nielsen equivalence in mapping tori over the torus

We use the geometry of the Farey graph to give an alternative proof of the fact that if A ∈ G L 2 Z A \in GL_2\mathbb {Z} and if G A = Z 2 ⋊ A Z G_A=\mathbb {Z}^2 \rtimes _A \mathbb {Z} is generated by two elements, then there is a single Nielsen equivalence class of 2 2 -element generating sets for G A G_A unless A A is conjugate to ± ( 2 a m p ; 1 1 a m p ; 1 ) \pm \left (\begin {smallmatrix} 2 & 1 \\ 1 & 1 \end {smallmatrix}\right ) , in which case there are two.

Cryptanalysis of a combinatorial public key cryptosystem

AbstractWe discuss pitfalls in the security of the combinatorial public key cryptosystem based on Nielsen transformations inspired by the ElGamal cryptosystem proposed by Fine, Moldenhauer and Rosenberger. We introduce three different types of attacks to possible combinatorial public key encryption schemes and apply these attacks to the scheme corresponding to the cryptosystem under discussion. As a result of our observation, we show that under some natural assumptions the scheme is vulnerable to at least one of the proposed attacks.

Nielsen equivalence in a class of random groups

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction.

Cryptographic Protocols Based on Nielsen Transformations

We introduce in this paper cryptographic protocols which use combinatorial group theory. Based on a combinatorial distribution of shares we present secret sharing schemes and cryptosystems using Nielsen transformations. Nielsen transformations are a linear technique to study free groups and general infinite groups. In addition the group of all automorphisms of a free group F, denoted by AUT (F), is generated by a regular Nielsen transformation between two basis of F, and each regular Nielsen transformation between two basis of F defines an automorphism of F.

Andrews–Curtis and Nielsen equivalence relations on some infinite groups

Abstract The Andrews–Curtis conjecture asserts that, for a free group Fn of rank n and a free basis (x 1,...,xn ), any normally generating tuple (y 1,...,yn ) is Andrews–Curtis equivalent to (x 1,...,xn ). This equivalence corresponds to the actions of Aut Fn and of F n n on normally generating n-tuples. The equivalence corresponding to the action of Aut Fn on generating n-tuples is called Nielsen equivalence. The conjecture for arbitrary finitely generated groups has its own importance to analyse potential counter-examples to the original conjecture. We study the Andrews–Curtis and Nielsen equivalence in the class of finitely generated groups for which every maximal subgroup is normal, including nilpotent groups and Grigorchuk groups.

Generating pairs and group actions

An action of a finite group on a closed 2-manifold is called almost free if it has a single orbit of points with nontrivial stabilizers. It is called large when the order of the group is greater than or equal to the genus of the surface. We prove that the orientation-preserving large almost free actions of G on closed orientable surfaces correspond to the Nielsen equivalence classes of generating pairs of G. We classify the almost free actions on the surfaces of genera 3 and 4, find the large almost free actions of the alternating group A5, and give various other examples.

NIELSEN EQUIVALENCE OF GENERATING PAIRS OF SL(2,q)

AbstractWe present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

FOLDING FREE-GROUP AUTOMORPHISMS

We describe an algorithm that uses Stallings' folding technique to decompose an element of Aut(Fn) as a product of Whitehead automorphisms (and hence as a product of Nielsen transformations). This algorithm is known to experts, but has not yet appeared in the literature. We use the algorithm to give an alternative method of finding a finite generating set for the subgroup of Aut(Fn) that fixes a subset Y of the basis elements, and the subgroup that fixes each element of Y up to conjugacy. We show that the intersection of this latter subgroup with I An is also finitely generated.

Rank of mapping tori and companion matrices

Given an element \varphi\in \mathrm {GL}(d,\mathbb Z) , consider the mapping torus defined as the semidirect product G=\mathbb Z^d\rtimes_\varphi\mathbb Z . We show that one can decide whether G has rank 2 or not (i.e.\ whether G may be generated by two elements). When G is 2-generated, one may classify generating pairs up to Nielsen equivalence. If \varphi has infinite order, we show that the rank of \mathbb Z^d\rtimes_{\varphi^n}\mathbb Z is at least 3 for all n large enough; equivalently, \varphi^n is not conjugate to a companion matrix in GL (d,\mathbb Z) if n is large.

Sets of primitive elements in a free group

A set of elements in a free group F is said to be a primitive set if it is a subset of some basis of F. In this paper several theorems about primitive sets are proven. The results are applications of Whiteheadʼs 3-dimensional model for studying automorphism of free groups; Nielsen transformations; and, the folding method of labeled graphs initiated by J. Stallings.

Nielsen equivalence classes of free abelianized extensions of groups

Let Open image in new window be a presentation of a group G. We obtain information about the Nielsen equivalence classes of Fn/R′ in terms of the group K1(ℤG) that arises in algebraic K-theory. We deduce some results about Nielsen equivalence classes and T-systems of polycyclic groups.

Mapping class factorization via fatgraph Nielsen reduction

The mapping class group of a genus $g$ surface $\Sigma_{g,1}$ with one boundary component is known to have a simple yet infinite presentation with generators given by elementary Whitehead moves on marked bordered fatgraphs. In this paper, we introduce an algorithm called fatgraph Nielsen reduction which, from the action of a mapping class $\varphi \in MC_{g,1}$ of $\Sigma_{g,1}$ on the fundamental group $\pi_{1}(\Sigma_{g,1})$ of $\Sigma_{g,1}$, determines a sequence of Whitehead moves representing $\varphi$ beginning at any choice of marked bordered fatgraph. The algorithm utilizes a reduction of bordered fatgraphs to linear chord diagrams, where the desired sequence is given in terms of elementary chord slide moves which continuously decrease some energy function. As a consequence, this leads to an algorithm which factors any mapping class given by its action on $\pi(\Sigma_{g,1})$ in terms of any convenient generating set for $MC_{g,1}$.