Nielsen Equivalence Research Articles

Extended Nielsen transformations and triviality of a group

Extended Nielsen transformations and triviality of a group

The Nielsen reduction and P-complete problems in free groups

The complexity of some classical algorithmic problems in free groups is studied. Problems like the generalized word problem, to determine shortest coset representatives, to decide whether two subgroups are equal or isomorphic, to decide whether a set of generators is independent or to decide whether a subgroup has finite index, are known to be decidable for free groups. We show that these problems are P-complete under log-space reducibility. This is proved by encoding the computations of a deterministic polynomially time bounded TM into a subgroup of a free group and implementing the Nielsen reduction, one of the main tools for solving algorithmic problems in free groups, in polynomial time.

Note on Automorphisms of a Free Abelian Group

Let F be a free group. Denote by the quotient group by the commutator subgroup which is a free abelian group. The fact that the natural map from Aut(F) into Aut() is an epimorphism, in case when F is finitely generated, was known as a consequence of the theory of Nielsen transformations ([2]) Proposition 4.4 and [3] Corollary 3.5.1).

On cancellations in HNN-groups

Nielsen [3] has given an algorithm for obtaining a free set of generators of a subgroup generated by arbitrary elements for free groups. In [10] Zieschang generalized this result to free products with amalgamation, and as an application he was able to calculate the rank of Fuchsian groups [-4, 10.1. More recently Rosenberger used these results in his work on Nielsen equivalence classes of certain groups [7, 8-1. Another approach to these problems was given by Lyndon and Schupp in [1]. In this paper we derive results analogous to those of Zieschang for HNN-groups by defining a suitable length function L and order relation for these groups. Our main results (Theorem (2.1) and (2.2)) may be summarized as follows: Let G = (H, t J rel H, t i K 1 t p~ K_ 1 ) be an HNN-extension of H and U c G be a system of elements of G which is minimal with respect to the defined order in G and free equivalence. Let w=ua ... un, u~'sU, ei= _+1, be an element of the subgroup generated by U such that L(w) < L(ui) for some i. Then there is a subword u~, ... u~s of w, conjugate to an element ofK,, /~ = + 1 and all the u~j lie in a subgroup conjugate to H. To illustrate our results we will give two short applications. First a new proof of a subgroup theorem of H.Neumann [2] (Theorem(3.1)) and secondly a new proof of a theorem of Pride [5] (Theorem (3.2)).

On the Generation of One-Relator Groups

This paper is concerned with obtaining information about the Nielsen equivalence classes and $T$-systems of certain two-generator HNN groups, and in particular of certain two-generator one-relator groups. The theorems presented here extend results of the author appearing in the Proceedings of the Second International Conference on the Theory of Groups. In particular it is shown here that if $G = \langle a,t;{a^{{\alpha _1}}}E_r^{ - 1}{a^{{\beta _1}}}{E_r} \cdots {a^{{a_s}}}E_r^{ - 1}{a^{{B_s}}}{E_r})$ where the ${\alpha _j}$ are positive, the ${\beta _i}$ are nonzero, ${E_r}$ has the form $[{a^{{ \in _1}}},[{a^{{ \in _2}}},[ \cdots ,[{a^{{ \in _r}}},t] \cdots ]]]$ with $|{ \in _1}| = |{ \in _2}| = \cdots = |{ \in _r}| = 1$, then in a large number of cases $G$ has one Nielsen equivalence class. Similar results are also obtained for certain groups with more than one relator. A fair proportion of the paper is given to developing a method for reducing pairs of elements in HNN groups. This method has some of the features of Nielsen’s reduction theorem for free groups. One other interesting result obtained here is that a one-relator group with torsion which has one $T$-system is Hopfian. The early part of the paper is discursive. It contains most of the known results concerning $T$-systems of one-relator groups, and highlights several open problems, some of which have been raised by other authors.

On the generation of one-relator groups

This paper is concerned with obtaining information about the Nielsen equivalence classes and T T -systems of certain two-generator HNN groups, and in particular of certain two-generator one-relator groups. The theorems presented here extend results of the author appearing in the Proceedings of the Second International Conference on the Theory of Groups. In particular it is shown here that if G = ⟨ a , t ; a α 1 E r − 1 a β 1 E r ⋯ a a s E r − 1 a B s E r ) G = \langle a,t;{a^{{\alpha _1}}}E_r^{ - 1}{a^{{\beta _1}}}{E_r} \cdots {a^{{a_s}}}E_r^{ - 1}{a^{{B_s}}}{E_r}) where the α j {\alpha _j} are positive, the β i {\beta _i} are nonzero, E r {E_r} has the form [ a ∈ 1 , [ a ∈ 2 , [ ⋯ , [ a ∈ r , t ] ⋯ ] ] ] [{a^{{ \in _1}}},[{a^{{ \in _2}}},[ \cdots ,[{a^{{ \in _r}}},t] \cdots ]]] with | ∈ 1 | = | ∈ 2 | = ⋯ = | ∈ r | = 1 |{ \in _1}| = |{ \in _2}| = \cdots = |{ \in _r}| = 1 , then in a large number of cases G G has one Nielsen equivalence class. Similar results are also obtained for certain groups with more than one relator. A fair proportion of the paper is given to developing a method for reducing pairs of elements in HNN groups. This method has some of the features of Nielsen’s reduction theorem for free groups. One other interesting result obtained here is that a one-relator group with torsion which has one T T -system is Hopfian. The early part of the paper is discursive. It contains most of the known results concerning T T -systems of one-relator groups, and highlights several open problems, some of which have been raised by other authors.

Classes of automorphisms of free groups of infinite rank

This paper is concerned with finding classes of automorphisms of an infinitely generated free group F which can be generated by “elementary” Nielsen transformations. Two different notions of “elementary” Nielsen transformations are explored. One leads to a classification of the automorphisms generated by these transformations. The other notion leads to the subgroup B of Aut ( F ) {\operatorname {Aut}}(F) consisting of the “bounded length” automorphisms of F. We prove that the class of “bounded 3-length” automorphisms B 3 {B_3} and the class of “elementary simultaneous” Nielsen transformations generate the same subgroup of Aut ( F ) {\operatorname {Aut}}(F) . We show that for the class T of automorphisms of “2 occurring generators", the groups generated by T ∩ B T \cap B and the “elementary simultaneous” Nielsen transformations are identical. These results lead to the conjecture that B is generated by the “elementary simultaneous Nielsen transformations". A study is also made of the subgroup S of the “triangular automorphisms” of F ∞ {F_\infty } , the free group on a countably infinite set of free generators. It is found that a “triangular automorphism” may be factored into “splitting automorphisms” of F ∞ {F_\infty } , which may be viewed as the “elementary” automorphisms of S.

Real Length Functions in Groups

This paper is a study of the structure of a group $G$ equipped with a ’length’ function from $G$ to the nonnegative real numbers. The properties that we require this function to satisfy are derived from Lyndon’s work on groups with integer-valued functions. A real length function is a function which assigns to each $g \in G$ a nonnegative real number $|g|$ such that the following axioms are satisfied: $|x| < |xx|$ if $x \ne 1$. $|x| = 0$ if and only if $x = 1$. $|{x^{ - 1}}| = |x|$. $c(x,y) \geq 0$ where $c(x,y) = 1/2(|x| + |y| - |x y^{-1}|)$. $c(x,y) \geq m$ and $c(y,z) \geq m$ imply $c(x,z) \geq m$. In this paper structure theorems are obtained for the cases when $G$ is abelian and when $G$ can be generated by two elements. We first prove that if $G$ is abelian, then $G$ is isomorphic to a subgroup of the additive group of the real numbers. Then we introduce a reduction process based on a generalized notion of Nielsen transformation. We apply this reduction process to finite sets of elements of $G$. We prove that if $G$ can be generated by two elements, then $G$ is either free or abelian.

Note on Nielsen transformations

Let al, , a. be free generators of the free group Fn, cl, cn those of the free group Fn, N an isomorphism of Fn onto Fn,t, and bi = N(a,) the maps of the a, under N. Then the bi are generators not only of Fn' but of any quotient group G'. I shall say that the generators bi of G' arise from the a, by a Nielsen transformation, N. In case Fn = Fn,tX N is (Nielsen) automorphism of Fn but need not be an automorphism of G = G'; nor is every automorphism of G expressible in terms of some N acting on a given set of generators. However, a given automorphism of G can be written as a Nielsen transformation acting on a suitably large set of generators (Theorem 2). I shall discuss this and related matters in the present paper. The following definitions and notation will be used. G= I{a,, * * *, an; R1, * * *, Rk} = {a; R} will denote the group given on n generators a, and k generating relations