Narrow Results

Let $FG$ be the group algebra of a residually torsion free nilpotent group $G$ over a field $F$ of characteristic $0$. If $(x,y)$ is any pair of noncommuting elements of $G$, we show that for any rational number $r$ with $r\ne 0,\pm 1$, the subgroup $〈1+rx,1+ry,1+rxy〉$ is free of rank $3$ in the Malcev–Neumann field of fractions of $FG$. This result comes from a method of producing free groups of rank three in rational quaternions. In general, if a division ring $D$ has dimension $d^2$ over its center $Z$, and if the transcendence degree of $Z$ over its prime field ${𝔽}_p$ is $\ge d^2+3$, we present a method to construct free subgroups of rank three.

Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → ℂ be a function for which ψ(x, y) only depends on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X × X. Moreover, we find a closed expression for the Schur norm ||ψ||_S of ψ. As applications, we obtaina closed expression for the completely bounded Fourier multiplier norm ||⋅||_M0A(G) of the radial functions on the free (non-abelian) group 𝔽_N on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the q-adic group PGL₂(ℚ_q) for every prime number q.

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓ_p. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

A complex number $\lambda$ is said to be non-free if the subgroup of $\mathrm{\text{SL}}(2,ℂ)$ generated by

Let F be the free group of rank 2, generated by x and y, and let w(x, y) ∈ F′ be a non-trivial word. We give elementary algebraic proofs and algorithms to (1) express [x, y]ⁿ as a product of [n/2] + 1 commutators and show this is the best possible; (2) show that (w(x, y))² cannot be written as one w-word and if g ≠ 1 ∈ w(F) then show that the minimal number of w-words required to express gⁿ as their product tends to infinity with n. Other results for free groups of higher rank are also presented.

Lyndon's group F^Z[x] is the free exponential group over the ring of integral polynomials Z[x]. This group, introduced by Lyndon in the 1960s, continues to be of interest to group theorists due to its importance in the study of first-order properties of free groups, in particular, equations over free groups. One of the crucial results of Lyndon's study was that the group F^Z[x] is fully residually F; i.e. for any finite collection of nontrivial elements in F^Z[x] there exists a homomorphism φ : F^Z[x] → F which is the identity on F and maps the given elements of F^Z[x] into nontrivial elements of F. The importance of F^Z[x] was further emphasized when Kharlampovich and Myasnikov proved in [3] that a finitely generated group is fully residually free if and only if it is embeddable into F^Z[x]. Lyndon's group and its subgroups play a vital role in the technique employed by O. Kharlampovich and A. Myasnikov in their solution of the famous Tarski problem on the decidability of the elementary theory of a free group (see [4, 5]).

In this paper, we show that Lyndon's group is conjugately residually free, i.e. it is possible to map F^Z[x] to the free group F preserving the nonconjugacy of two elements. This result is a further step towards the understanding of the properties of F^Z[x]; moreover, it is closely related to the problem of "lifting solutions" of equations from F to F^Z[x], since our result implies that the solutions can indeed be "lifted" from F to F^Z[x] for equations of the type x^-1 c₁ x = c₂.

The structure of Lyndon's group, described by A. Myasnikov and V. Remeslennikov in [8], involves an infinite sequence of free constructions of a specific type, called free extensions of centralizers. For more results on residual properties of certain types of free constructions, see also the works of Ribes, Segal and Zalesskii (for example [9]).

We introduce a family of multiplicative distributions {μ_s|s∈(0,1)} on a free group F and study it as a whole. In this approach, the measure of a given set R⊆F is a function μ(R) : s → μ_s(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.

This paper deals with the study of the free noncommutative group in the multiplicative group of the skewfield of the real Hamilton quaternions. The main results proved in this paper allows us to obtain the following interesting corollary: let G be a subgroup of rational quaternions. Then G is either solvable or contains the free noncommutative group.

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception.

The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

We give an asymptotic estimate for the growth of prime conjugacy classes in a finitely generated free group.

We prove a constructive version of Tits' alternative for groups of quaternions with algebraic coefficients by bounding valuations of their entries considered as elements of a fraction field of an opportunely chosen Dedekind domain.

Stalling's folding process is a key algorithm for solving algorithmic problems for finitely generated subgroups of free groups. Given a subgroup H = 〈J₁,…,J_m〉 of a finitely generated nonabelian free group F = F(x₁,…,x_n) the folding porcess enables one, for example, to solve the membership problem or compute the index [F : H]. We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stallings' folding process in time O(N log*(N)), where N is the sum of the word lengths of the given generators of H.

We show that the following problems are decidable in a rank 2 free group F₂: Does a given finitely generated subgroup H 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 if we allow H to be a rational subset of F₂, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, the following weaker problem is decidable: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.

We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

We develop a theory of generalized presentations of groups and give a generalized presentation of the automorphism group of the free group of countable rank, Aut(F_ω).

We say a subset Σ ⊆ F_N of the free group of rank N is spectrally rigid if whenever T₁, T₂ ∈ cv_N are ℝ-trees in (unprojectivized) outer space for which ‖σ‖_T1 = ‖σ‖_T2 for every σ ∈ Σ, then T₁ = T₂ in cv_N. The general theory of (non-abelian) actions of groups on ℝ-trees establishes that T ∈ cv_N is uniquely determined by its translation length function ‖⋅‖_T : F_N → ℝ, and consequently that F_N itself is spectrally rigid. Results of Smillie and Vogtmann, and of Cohen, Lustig and Steiner establish that no finite Σ is spectrally rigid. Capitalizing on their constructions, we prove that for any Φ ∈ Aut(F_N) and g ∈ F_N, the set Σ = {Φⁿ(g)}_n∈ℤ is not spectrally rigid.

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety A_mA_n, and let A = {a₁,…, a_r} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {a_l+1,…, a_r} for some l < r, then S is a subset of a basis for the relatively free group on {a₁,…, a_l}.

We study free subgroups in the multiplicative group of division rings generated by group rings of some classes of soluble groups.

It is shown that the big free group (the set of countably-long words over a countable alphabet) is almost free, in the sense that any function from the alphabet to a compact topological group factors through a homomorphism. This statement is in fact a simple corollary of the more general result proven below on the extendability of homomorphisms from subgroups (of a certain kind) of the big free group to a compact topological group.