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On the complexity of the Whitehead minimization problem
 PREPRINT 721, CENTRE DE RECERCA MATEMÀTICA
, 2006
"... 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 b ..."
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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.
Algebraic extensions in free groups
 TRENDS MATH., BIRKHÄUSER
, 2007
"... 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 grou ..."
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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 fieldtheoretic 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.
Primitive words, free factors and measure preservation, Arxiv preprint arXiv:1104.3991
, 2011
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Automorphic orbits in free groups: words versus subgroups ∗
, 2009
"... 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 even if we restrict ..."
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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 even 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 kalmost bounded automorphism? An automorphism is kalmost bounded if at most one of the letters has an image of length greater than k.
Generic properties of random subgroups of a free group for general distributions
"... We consider a generalization of the uniform wordbased distribution for finitely generated subgroups of a free group. In our setting, the number of generators is not fixed, the length of each generator is determined by a random variable with some simple constraints and the distribution of words of a ..."
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We consider a generalization of the uniform wordbased distribution for finitely generated subgroups of a free group. In our setting, the number of generators is not fixed, the length of each generator is determined by a random variable with some simple constraints and the distribution of words of a fixed length is specified by a Markov process. We show by probabilistic arguments that under rather relaxed assumptions, the good properties of the uniform wordbased distribution are preserved: generically (but maybe not exponentially generically), the tuple we pick is a basis of the subgroup it generates, this subgroup is malnormal and the group presentation defined by this tuple satisfies a small cancellation condition.
Centro de Matemática, Universidade do Porto †
"... On finiteindex extensions of subgroups of free groups ∗ ..."
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On finiteindex extensions of subgroups of free groups ∗
, 2009
"... We study the lattice of finiteindex extensions of a given finitely generated subgroup H of a free group F. This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of H. This characterization leads to a fast algorithm to compute the com ..."
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We study the lattice of finiteindex extensions of a given finitely generated subgroup H of a free group F. This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of H. This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a subexponential and superpolynomial upper bound for the number of finiteindex extensions of H, and we give a languagetheoretic characterization of the lattice of finiteindex subgroups of H. Finally, we give a polynomial time algorithm to compute the malnormal closure of H.