We show that the following problems are decidable in a rank 2 free group F2: does a given finitely generated subgroupH contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of automorphisms of F2? Moreover, decidability subsists even if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms)....

AbstractLet W be a non-empty subset of a free group. The automorphism of a group G is said to be a marginal automorphism, if for all x in G,x^−1alpha(x) in W^*(G), where W^*(G) is the marginal subgroup of G.In this paper, we give necessary and sufficient condition for a purelynon-abelian p-group G, such that the set of all marginal automorphismsof G forms an elementary abelian p-group.

Let G be a group generated by r elements g1, g2, . . . , gr. Among the reduced words in g1, g2, . . . , gr of length n some, say γn, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of γ2n has a limit, called the cogrowth exponent with respect to generators g1, g2, . . . , gr. We show by analytic methods that the numbers γn vary regularl...

We prove the existence of a free group of rotations of rank 2 which acts on the rational unit sphere without non-trivial fixed points. Introduction. The purpose of this paper is to prove that the group SO3(Q) of all proper orthogonal 3 × 3 matrices with rational entries has a free subgroup F2 of rank 2 such that for all w ∈ F2 different from the identity and for all ~r ∈ S2 ∩Q3 we have w(~r ) 6...

Let Jl consist of the groups GÍL m) = (a, b; a~ b a = b"1) where |/.| 4 1 4 |zzi|, lm40 and l, m are coprime. We characterize the endomorphisms of these groups, compute the centralizers of special elements and show that the endomorphism a —• a, b —' b is onto with a nontrivial fully invariant kernel. Hence G(l, m) is non-Hopfian in the'fully invariant sense.' Our purpose is to prove the results...

Let g, p ∈ [0↑∞ [, the set of non-negative integers. Let Ag,p denote the group consisting of all those automorphisms of the free group on t[1↑p]∪x[1↑g]∪y[1↑g] which fix the element Π j∈[p↓1] tj Π i∈[1↑g] [xi, yi] and permute the set of conjugacy classes { [tj ] : j ∈ [1↑p]}. Labruère and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others...

We find a lower bound to the size of finite groups detecting a given word in the free group. More precisely we construct a word wn of length n in non-abelian free groups with the property that wn is the identity on all finite quotients of size ∼ n2/3 or less. This improves on a previous result of BouRabee and McReynolds quantifying the lower bound of the residual finiteness of free groups. A gr...

We exhibit normal subgroups of a free nilpotent group F of rank two and class three, which have isomorphic finite quotients but are not conjugate under any automorphism of F . A remarkable fact about free profinite groups of finite rank is that any isomorphism between finite quotients of such a group F lifts to an automorphism of F . This is true, more generally, if F a free pro-Cgroup of finit...

A Thue system T over ~ is said to allow nontrivial elements of finite order, if there exist a word u E ~* and integers n ;;. 0 and k ;;. 1 such that u .... fA and u,,+k .... f u". Here the following decision problem is shown to be decidable: Instance. A finite, monadic, Church-Rosser Thue system Tover~. Question. Does T allow nontrivial elements of finite order? By a result of Muller and Schupp...

In this paper we show that several classes of elementary properties (properties definable by sentences of a first order logic) of groups hold for all nonabelian free groups. These results are obtained by examining special embeddings of these groups into one another which preserve the properties in question. I. It has long been conjectured that the finitely generated nonabelian free groups are e...