On the fourth Johnson homomorphism of the automorphism group of a free group

In this paper we consider the Johnson homomorphism of the automorphism group of a free group with respect to the lower central series of the IA-automorphism group of a free group. In particular, we determine the rational cokernel of the fourth Johnson homomorphism, and show that there appears a new obstruction for the surjectivity of the Johnson homomorphism. Furthermore we characterize this obstruction using trace maps.     

On the intersection of subgroups of a free group

Summary The Hanna Neumann Conjecture says that the intersection of subgroups of rankn+1 andm+1 of a free group has rank at mostnm+1. This paper proves the conjecture for the casem=1. (See Theorem 1.) Our methods imply that the strengthened Hanna Neumann Conjecture is also true in this case (Theorem 2).Oblatum 31-V-1991 & 8-X-1991     

Finite quotients of the automorphism group of a free group

The automorphism group AutF_n of a free group F_n of rank n acts on the product of n copies of a group G by substituting n elements of G into the words defining an automorphism of the free group. This gives rise to an antihomomorphism from AutF_nto a permutation group. We determine this antihomomorphic image of AutF_n when G is the semidirect product Z_p x Z_q     

Strongly dense free subgroups of semisimple algebraic groups

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type     

A presentation for the special automorphism group of a free group

A presentation is given for SA_n, the group of automorphisms of determinant 1 of a free group F_n of rank n. The canonical isomorphisms $\text{H}{}_2\text{(A}{}_{\mathrm{n}}\text{,}\text{Z}\text{)}\text{?}\text{H}{}_2\text{(SA}{}_{\mathrm{n}}\text{,}\text{Z}\text{)}\text{?}\text{K}{}_2\text{(}\text{Z}\text{)}$ are established for n ≥ 5, where A_n is the full group of automorphisms of F_n.     

On maximal proper subgroups of field automorphism groups

Let G be the automorphism group of an extension of algebraically closed fields of characteristic zero of transcendence degree n, 1 ≤ n ≤ ∞. In this paper we

Twisted second homology groups of the automorphism group of a free group

In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for the 2-torsion part, using combinatorial group theory.     

Twisted first homology groups of the automorphism group of a free group

The automorphism group and outer automorphism group of a free group F_n of rank n act on the abelianized group H of F_n and the dual group H^* of H. The twisted first homology groups of and with coefficients in H and H^* are calculated.     

On the outer automorphism group of a hyperbolic group

LetG be a one-ended, word-hyperbolic group. Let Γ be an irreducible lattice in a connected semi-simple Lie group of rank at least 2. Ifh: Γ→Out(G) is a homomorphism, then Im(h) is finite. Dedicated to Professor Takushiro Ochiai for his sixtieth birthday     

On the automorphism group of a tree

It is shown that H = Γ(T)_v is normal in G = Γ(T_v) for any tree T and any vertex v, if and only if, for all vertices u in the neighborhood N of v, the set of images of u under G is either contained in N or has precisely the vertex u in common with N and every vertex in the set of images is fixed by H. Further, if S is the smallest normal subgroup of G containing H then $\text{G}\text{S}$ is the direct product of the wreath products of various symmetric groups around groups of order 1 or 2. The degrees of the symmetric groups involved depend on the numbers of isomorphic components of T_v and the structure of such components.     

On the automorphism group of a cone

Let V be a real finite dimensional vector space, and let C be a full cone in C. In Sec. 3 we show that the group of automorphisms of a compact convex subset of V is compact in the uniform topology, and relate the group of automorphisms of C to the group of automorphisms of a compact convex cross-section of C. This section concludes with an application which generalizes the result that a proper Lorentz transformation has an eigenvector in the light cone. In Sec. 4 we relate the automorphism group of C to that of its irreducible components. In Sec. 5 we show that every compact group of automorphisms of C leaves a compact convex cross-section invariant. This result is applied to show that if C is a full polyhedral cone, then the automorphism group of C is the semidirect product of the (finite) automorphism group of a polytopal cross-section and a vector group whose dimension is equal to the number of irreducible components of C. An example shows that no such result holds for more general cones.