Enric Ventura

The Group Fixed by a Family of Injective Endomorphisms of a Free Group

Groupoids Measuring devices Properties of the basic operations Minimal representatives and fixed subgroupoids Open problems Bibliography Index.

Orbit decidability and the conjugacy problem for some extensions of groups

Given a short exact sequence of groups with certain conditions, 1 ? F ? G ? H ? 1, we prove that G has solvable conjugacy problem if and only if the corresponding action subgroup A 6 Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z2?Fm, F2?Fm, Fn?Z, and Zn?A Fm with virtually solvable action group A 6 GLn(Z). Also, we give an easy way of constructing groups of the form Z4?Fn and F3?Fn with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in Aut(F2) is given.

Algebraic Extensions in Free Groups

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical field-theoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g., being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.

ON THE COMPLEXITY OF THE WHITEHEAD MINIMIZATION PROBLEM

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem — to decide whether a word is an element of some basis of the free group — and the free factor problem can also be solved in polynomial time.

Absolute-Type Shaft Encoding Using LFSR Sequences With a Prescribed Length

Maximal-length binary sequences have existed for a long time. They have many interesting properties, and one of them is that, when taken in blocks of n consecutive positions, they form 2n - 1 different codes in a closed circular sequence. This property can be used to measure absolute angular positions as the circle can be divided into as many parts as different codes can be retrieved. This paper describes how a closed binary sequence with an arbitrary length can be effectively designed with the minimal possible block length using linear feedback shift registers. Such sequences can be used to measure a specified exact number of angular positions using the minimal possible number of sensors that linear methods allow.

On fixed subgroups of maximal rank

We show that, in the free group F of rank n, n is the maximal length of strictly ascending chains of maximal rank fixed subgroups, that is, rank n subgroups of the form Fix^ for some 4> L Aut(F). We further show that, in the rank two case, if the intersection of an arbitrary family of proper maximal rank fixed subgroups has rank two then all those subgroups are equal. In particular, every G < Aut(F) with r(FixG) = 2 is either trivial or infinite cyclic.

ON ENDOMORPHISMS OF TORSION-FREE HYPERBOLIC GROUPS

Let H be a torsion-free δ-hyperbolic group with respect to a finite generating set S. From the main result in the paper, Theorem 1.2, we deduce the following two corollaries. First, we show that there exists a computable constant such that, for any endomorphism φ of H, if φ(h) is conjugate to h for every element h ∈ H of length up to , then φ is an inner automorphism. Second, we show a mixed (conjugate/non-conjugate) version of the classical Whitehead problem for tuples is solvable in torsion-free hyperbolic groups.

Irreducible automorphisms of growth rate one

In proving the conjecture of Scott that the rank of the fixed group of an automorphism of a finitely generated free group F is at most the rank of F, Bestvina and Handel [l] use a hierarchy for the set of automorphisms of finitely generated free groups and, from their point of view, the irreducible automorphisms of growth rate 1 are the simplest. It is not immediately obvious from [l] what these automorphisms are, and our purpose here is to describe them explicitly. We begin with a vocabulary review.

Fixed Subgroups are Compressed in Free Groups

Abstract In this paper, we prove that the fixed subgroup of an arbitrary family of endomorphisms ψ i , i ∈ I, of a finitely generated free group F, is F-super-compressed. In particular, r(∩ i∈I Fix ψ i ) ≤ r(M) for every subgroup M ≤ F containing ∩ i∈I Fix ψ i . This provides new evidence towards the inertia conjecture for fixed subgroups of free groups. As a corollary, we show that, in the free group of rank n, every strictly ascending chain of fixed subgroups has length at most 2n. This answers a question of Levitt.

Finite automata for Schreier graphs of virtually free groups

Abstract The Stallings construction for f.g. subgroups of free groups is generalized by introducing the concept of Stallings section, which allows efficient computation of the core of a Schreier graph based on edge folding. It is proved that the groups that admit Stallings sections are precisely the f.g. virtually free groups, this is proved through a constructive approach based on Bass–Serre theory. Complexity issues and applications are also discussed.