On dense free subgroups of the automorphism group

We obtain results on existence of dense free subgroups of the automorphism group of a homogeneous model.     

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.     

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     

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 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 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 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.     

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 centralizer and the commutator subgroup of an automorphism

Let φ be an automorphism of a group G. In this paper, we study the influence of its centralizer on its commutator subgroup when G is polycyclic or metabelian. For instance, when G is metabelian and φ fixed-point-free of prime order p, we prove that is nilpotent of class ≤ p. Also, when G is polycyclic and φ of order 2, we show that if is finite, then so are and .     

On the centralizer and the commutator subgroup of an automorphism

Let φ be an automorphism of a group G. In this paper, we study the influence of its centralizer \({C_G(\varphi)}\) on its commutator subgroup \({[G,\varphi]}\) when G is polycyclic or metabelian. For instance, when G is metabelian and φ fixed-point-free of prime order p, we prove that \({[G,\varphi]}\) is nilpotent of class ≤ p. Also, when G is polycyclic and φ of order 2, we show that if \({C_G(\varphi)}\) is finite, then so are \({G/[G,\varphi]}\) and \({[G,\varphi]'}\) .     

On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ _p wrZ _p , almost all Cayley graphs ofG have automorphism group isomorphic toG.