In this paper we study obstructions to presentability by products for finitely generated groups. Along the way we develop both the concept of acentral subgroups, and the relations between presentability by products on the one hand, and certain geometric and measure or orbit equivalence invariants of groups on the other. This leads to many new examples of groups not presentable by products, incl...

We prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups. We also prove that it is decidable whether or not a finitely generated submonoid of a free group is graded, and solve the homomorphism and isomorphism problems for graded submonoids of free groups. This generalizes earlier results for submonoids of free monoids. MSC: 20E05; 20...

Let G be any locally compact metrizable group. The main result of this paper, roughly stated, is that if F < G is any finitely generated free group and Γ < G any lattice, then up to a small perturbation and passing to a finite index subgroup, F is a subgroup of Γ. If G/Γ is noncompact then we require additional hypotheses that include G = SO(n, 1). MSC: 20E07, 20F65, 20F67, 22D40, 20E05.

Elements of the commutator subgroup of a free group F can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus g, there is a sequence of words wg which can be presented by f .g/ distinct Wicks forms, where f .g/ > gŠ. Moreover we may choose these words wg to be square-free. Mathematics Subject Classification (2000). 20E05, 20F10.

Let V be a pseudovariety of finite groups such that free groups are residually V, and let φ:F (A) → F (B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension φ̂ of φ between the pro-V completions of F (A) and F (B) is also injective. In particular, if V is extension-closed, this is the case if and only if φ(F (A)) and its...

We show that if a group G acting faithfully on a rooted tree T has a free subgroup, then either there exists a point w of the boundary @T and a free subgroup of G with trivial stabilizer ofw, or there existsw 2 @T and a free subgroup ofG fixingw and acting faithfully on arbitrarily small neighborhoods of w. This can be used to prove the absence of free subgroups for different known classes of g...

In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they accept. We show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoids. These automata allow us to compute explicitly generating functions, λ−measures and Cesaro meas...

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of ...

The Andrews-Curtis conjecture states that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of the elementary Nielsen transformations and conjugations. In this paper we describe all balanced presentations of the trivial group on two generators and with the total length of relators ≤ 12. We show that all these presentations satisfy the Andrews-Curt...

The probability that a random pair of elements from the alternating group An generates all of An is shown to have an asymptotic expansion of the form 1¡1/n¡ 1/n2¡4/n3¡23/n4¡171/n5¡ ... . This same asymptotic expansion is valid for the probability that a random pair of elements from the symmetric group Sn generates either An or Sn. Similar results hold for the case of r generators (r > 2). MSC 2...