On a Problem of Fox.
On an algorithm to decide whether a free group is a free factor of another
We revisit the problem of deciding whether a finitely generated subgroup is a free factor of a given free group . Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of and exponential in the rank of . We show that the latter dependency can be made exponential in the rank difference rank - rank, which often makes a significant change.
On an algorithm to decide whether a free group is a free factor of another
We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in the rank of F. We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change.
On D. I. Moldavanskij's question on -separable subgroups of a free group.
On exponential growth rates for free groups.
Let Fp be a free group of rank p ≥ 2. It is well-known that, with respect to a p-element generating set, that is, a basis, the exponential growth rate of Fp is 2p-1. We show that the exponential growth rate τ of a group G with respect to a p-element generating set X is 2p-1 if and only if G is free on X; otherwise τ < 2p-1. We also prove that, for any finite generating set X of Fp which is disjoint from X-1, the exponential growth rate τ of Fp with respect to X is 2p-1 if and only if X is...
On free subgroups of units in quaternion algebras
It is well known that for the ring H(ℤ) of integral quaternions the unit group U(H(ℤ) is finite. On the other hand, for the rational quaternion algebra H(ℚ), its unit group is infinite and even contains a nontrivial free subgroup. In this note (see Theorem 1.5 and Corollary 2.6) we find all intermediate rings ℤ ⊂ A ⊆ ℚ such that the group of units U(H(A)) of quaternions over A contains a nontrivial free subgroup. In each case we indicate such a subgroup explicitly. We do our best to keep the arguments...
On free subgroups of units in quaternion algebras II
Let A ⊆ ℚ be any subring. We extend our earlier results on unit groups of the standard quaternion algebra H(A) to units of certain rings of generalized quaternions H(A,a,b) = ((-a,-b)/A), where a,b ∈ A. Next we show that there is an algebra embedding of the ring H(A,a,b) into the algebra of standard Cayley numbers over A. Using this embedding we answer a question asked in the first part of this paper.
On Gupta Representations of Central Extensions.
On infinite Nielsen transformations.
On some free semigroups, generated by matrices
Let We call a complex number “semigroup free“ if the semigroup generated by and is free and “free” if the group generated by and is free. First families of semigroup free ’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free ’s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common picture....
On some systems of equations with constraints in a free group. – Addenda. (Sur certains systèmes d'équations avec constraintes dans un groupe libre. – Addenda.)
On some systems of equations with constraints in a free group. (Sur certains systèmes d'équations avec contraintes dans un groupe libre.)
On strictly sparse subsets of a free group.
On tame automorphisms of some metabelian groups.
On the combinatorics of permutations. (Zur Kombinatorik von Permutationen.)
On the dynamics and the fixed subgroup of a free group automorphism.
On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane
We prove several results concerning the existence of universal covering spaces for separable metric spaces. To begin, we define several homotopy-theoretic conditions which we then prove are equivalent to the existence of a universal covering space. We use these equivalences to prove that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal covering space. As an application of these results, we prove the main result of this article,...
On the intersection of finitely generated subgroups of free groups.
On the intersection of subgroups of a free group.
On the inverse limit of free nilpotent groups.