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 ...

For any finitely generated group G, two complexity functions αG and βG are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of G and the norm of its inverse. Restricting attention to free groups, Fr, the exact asymptotic behaviour of α2 and β2 is computed. For rank r > 3, polynomial lower bounds are provided for αr and βr, and the...

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...

A generalised tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalised triangle groups and whose edge groups are finite cyclic. We prove an improved Spelling Theorem for generalised triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalised tetrahedron groups ac...

Let w be a non-trivial word in two variables. We prove that the probability that two randomly chosen elements x, y of a nonabelian finite simple group S satisfy w(x, y) = 1 tends to 0 as |S| → ∞. As a consequence, we obtain a new short proof of a well-known conjecture of Magnus concerning free groups, as well as some applications to profinite groups. Research partially supported by NSERC grant ...

A self-map, either single-valued or multi-valued, is multiply fixed if every map homotopic to it has at least two fixed points. Let p : X̃ → X be a finite covering space, of degree n, of a connected finite polyhedron, and let f : X → X be a map. We lift f to an n-valued map φp,f : X̃ ( X̃ and prove that it is multiply fixed if the Nielsen number of f is greater than or equal to two. We obtain a fo...

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F2) of automorphisms of the rank two free group F2 and show that it can be realized as a monoid in the group B4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F2 lifting any given basis of the free abelian group Z . ...

This article surveys many standard results about the braid group, with emphasis on simplifying the usual algebraic proofs. We use van der Waerden’s trick to illuminate the Artin-Magnus proof of the classic presentation of the braid group considered as the algebraic mapping-class group of a disc with punctures. We give a simple, new proof of the σ1-trichotomy for the braid group, and, hence, rec...