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ABSTRACT: This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations. CONTENTS

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 time-complexity 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.

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 hyper-surface in�9 mapping onto�8. Our choice of generators exhibit Out(F2) symmetries which allow for a succinct expression of the defining relations. 1.

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.

Dedicated to Professor Yukio Matsumoto on the occasion of his sixtieth birthday Abstract. A theorem of Kirby states that two framed links in the 3-sphere yield orientation-preserving homeomorphic results of surgery if these links are related by a sequence of stabilization and handle-slides. The purpose of the present paper is twofold: We give a sufficient condition for a sequence of handle-slides on framed links to be able to be replaced by a sequences of algebraically canceling pairs of handle-slides. Using this result, we obtain a refinements of Kirby’s calculus for integral homology spheres. 1.

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.

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 upper-bound 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.

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.

. We classify all complex representations of Aut Fn ; the automorphism group of the free group Fn (n 3); of dimension 2n \Gamma 2: Among those representations is a new representation of dimension n+1 which doesn't vanish on the group of inner automorphisms. Introduction This paper continues the study of low-dimensional linear representations of \Gamma n = Aut F n (n 3), the automorphism group of the free group, begun in [R1] (low-dimensional representations of Aut F 2 were analyzed in [DP]). It was shown in [R1] (cf. also Theorem 1.2 below) that any n-dimensional representation of Aut F n factors through the the canonical homomorphism f : \Gamma n ! GL n (Z): In this paper we will show that in dimension higher than n; the group \Gamma n acquires new representations. Namely, we will establish the existence of a homomorphism g : \Gamma n ! GL n (Z) n Z n which "lifts" f and gives rise to an (n + 1)-dimensional representation. The main result of the paper (Theorem 3.1) claims t...