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Automorphisms of free groups and outer space
 in “Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000)”, Geom. Dedicata 94
, 2002
"... ABSTRACT: This is a survey of recent results in the theory of automorphism groups of finitelygenerated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations. CONTENTS ..."
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Cited by 17 (5 self)
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ABSTRACT: This is a survey of recent results in the theory of automorphism groups of finitelygenerated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations. CONTENTS
Whitehead Method and Genetic Algorithms
 Contemporary Mathematics
"... In this paper we discuss a genetic version (GWA) of the Whitehead's algorithm, which is one of the basic algorithms in combinatorial group theory. It turns out that GWA is surprisingly fast and outperforms the standard Whitehead's algorithm in free groups of rank 5. Experimenting with GWA we co ..."
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Cited by 9 (5 self)
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In this paper we discuss a genetic version (GWA) of the Whitehead's algorithm, which is one of the basic algorithms in combinatorial group theory. It turns out that GWA is surprisingly fast and outperforms the standard Whitehead's algorithm in free groups of rank 5. Experimenting with GWA we collected an interesting numerical data that clari es the timecomplexity of the Whitehead's Problem in general. These experiments led us to several mathematical conjectures. If con rmed they will shed light on hidden mechanisms of Whitehead Method and geometry of automorphic orbits in free groups.
GENERATORS, RELATIONS AND SYMMETRIES IN PAIRS OF 3 × 3 UNIMODULAR MATRICES
, 2007
"... Abstract. Denote the free group on two letters by F2 and the SL(3,�)representation variety of F2 by R = Hom(F2, SL(3,�)). There is a SL(3,�)action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine variety X. We determine explicit minimal generators and ..."
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Cited by 6 (4 self)
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Abstract. Denote the free group on two letters by F2 and the SL(3,�)representation variety of F2 by R = Hom(F2, SL(3,�)). There is a SL(3,�)action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine variety X. We determine explicit minimal generators and defining relations for the subring of invariants and show X is a degree 6 hypersurface in�9 mapping onto�8. Our choice of generators exhibit Out(F2) symmetries which allow for a succinct expression of the defining relations. 1.
A presentation for Aut(Fn
, 2006
"... Abstract. We study the action of the group Aut(Fn) of automorphisms of a finitely generated free group on the degree 2 subcomplex of the spine of Auter space. Hatcher and Vogtmann showed that this subcomplex is simply connected, and we use the method described by K. S. Brown to deduce a new presenta ..."
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Cited by 2 (1 self)
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Abstract. We study the action of the group Aut(Fn) of automorphisms of a finitely generated free group on the degree 2 subcomplex of the spine of Auter space. Hatcher and Vogtmann showed that this subcomplex is simply connected, and we use the method described by K. S. Brown to deduce a new presentation of Aut(Fn). 1.
REFINED KIRBY CALCULUS FOR INTEGRAL HOMOLOGY SPHERES
, 2005
"... Dedicated to Professor Yukio Matsumoto on the occasion of his sixtieth birthday Abstract. A theorem of Kirby states that two framed links in the 3sphere yield orientationpreserving homeomorphic results of surgery if these links are related by a sequence of stabilization and handleslides. The purp ..."
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Dedicated to Professor Yukio Matsumoto on the occasion of his sixtieth birthday Abstract. A theorem of Kirby states that two framed links in the 3sphere yield orientationpreserving homeomorphic results of surgery if these links are related by a sequence of stabilization and handleslides. The purpose of the present paper is twofold: We give a sufficient condition for a sequence of handleslides on framed links to be able to be replaced by a sequences of algebraically canceling pairs of handleslides. Using this result, we obtain a refinements of Kirby’s calculus for integral homology spheres. 1.
Dimension of the Torelli group for Out(Fn
 Invent. Math
"... Abstract. Let Tn be the kernel of the natural map Out(Fn) → GLn(Z). We use combinatorial Morse theory to prove that Tn has an Eilenberg–MacLane space which is (2n − 4)dimensional and that H2n−4(Tn, Z) is not finitely generated (n ≥ 3). In particular, this recovers the result of Krstić–McCool that ..."
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Cited by 2 (1 self)
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Abstract. Let Tn be the kernel of the natural map Out(Fn) → GLn(Z). We use combinatorial Morse theory to prove that Tn has an Eilenberg–MacLane space which is (2n − 4)dimensional and that H2n−4(Tn, Z) is not finitely generated (n ≥ 3). In particular, this recovers the result of Krstić–McCool that T3 is not finitely presented. We also give a new proof of the fact, due to Magnus, that Tn is finitely generated. 1.
ANOTHER VIEW OF THE GAUSSIAN ALGORITHM
"... Abstract. We introduce here a rewrite system in the group of unimodular matrices, i.e., matrices with integer entries and with determinant equal to ±1. We use this rewrite system to precisely characterize the mechanism of the Gaussian algorithm, that finds shortest vectors in a two–dimensional latti ..."
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Abstract. We introduce here a rewrite system in the group of unimodular matrices, i.e., matrices with integer entries and with determinant equal to ±1. We use this rewrite system to precisely characterize the mechanism of the Gaussian algorithm, that finds shortest vectors in a two–dimensional lattice given by any basis. Putting together the algorithmic of lattice reduction and the rewrite system theory, we propose a new worst–case analysis of the Gaussian algorithm. There is already an optimal worst–case bound for some variant of the Gaussian algorithm due to Vallée [16]. She used essentially geometric considerations. Our analysis generalizes her result to the case of the usual Gaussian algorithm. An interesting point in our work is its possible (but not easy) generalization to the same problem in higher dimensions, in order to exhibit a tight upperbound for the number of iterations of LLL–like reduction algorithms in the worst case. Moreover, our method seems to work for analyzing other families of algorithms. As an illustration, the analysis of sorting algorithms are briefly developed in the last section of the paper. 1.
THE DECATEGORIFICATION OF SUTURED FLOER HOMOLOGY
, 903
"... Abstract. We define a torsion invariant for balanced sutured manifolds and show that it agrees with the Euler characteristic of sutured Floer homology. The torsion is easily computed and shares many properties of the usual Alexander polynomial. 1. ..."
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Abstract. We define a torsion invariant for balanced sutured manifolds and show that it agrees with the Euler characteristic of sutured Floer homology. The torsion is easily computed and shares many properties of the usual Alexander polynomial. 1.
ORIENTATIONREVERSING FREE ACTIONS ON HANDLEBODIES
, 2004
"... Abstract. We examine free orientationreversing group actions on orientable handlebodies, and free actions on nonorientable handlebodies. A classification theorem is obtained, giving the equivalence classes and weak equivalence classes of free actions in terms of algebraic invariants that involve Ni ..."
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Abstract. We examine free orientationreversing group actions on orientable handlebodies, and free actions on nonorientable handlebodies. A classification theorem is obtained, giving the equivalence classes and weak equivalence classes of free actions in terms of algebraic invariants that involve Nielsen equivalence. This is applied to describe the sets of free actions in various cases, including a complete classification for many (and conjecturally all) cases above the minimum genus. For abelian groups, the free actions are classified for all genera. The orientationpreserving free actions of a finite group G on 3dimensional orientable handlebodies have a close connection with a longstudied concept from group theory, namely Nielsen equivalence of generating sets. The basic result is that the orientationpreserving free actions of G on the handlebody of genus g, up to equivalence, correspond to the Nielsen equivalence classes of nelement generating sets of G, where n = 1 + (g − 1)/G. This has been known for a long time; it is implicit in work of J. Kalliongis and A.