We introduce conditions on a group action on a tree that are su½cient for the action to extend to the automorphism group. We apply this to two di¨erent classes of one-relator groups: certain Baumslag±Solitar groups and one-relator groups with centre. In each case we derive results about the automorphism group, and deduce that there are relatively few outer automorphisms.

We investigate asphericity of the relative group presentation G, t | atbtctdtet = 1 and prove it aspherical provided the subgroup of G generated by {ab −1 , bc −1 , cd −1 , de −1 } is neither finite cyclic nor a finite triangle group. We also prove a similar result for the closely related relative group presentation G, s, t | αsβsγt = 1 = δtεtζs −1 .

In the present note we investigate asphericity (as defined in [2]) of the relative group presentation P = H, t|(ta) p (tb) q (tc) r = 1 with a, b, c ∈ H and p, q, r > 1. Results similar to those proved in [1, 2, 7] are obtained.

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Moreover, we investigate the profinite topology of F 2 × F 2 and we show that the profinite topology of the above group is strongly connected with the profinite topology of F 2 .

Given ϕ ∈ GL(d, Z), it is decidable whether the mapping torus G = Z d ⋊ ϕ Z has rank 2 or not (i.e. whether G may be generated by two elements); when it does, one may classify generating pairs up to Nielsen equivalence. If ϕ has infinite order, the rank of Z d ⋊ ϕ n Z is at least 3 for all n large enough; equivalently, ϕ n is not conjugate to a companion matrix in GL(d, Z) if n is large.

a b s t r a c tWe show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.

Abstract. We show that the holomorph of the free group on two generators satisfies the Farrell–Jones Fibered Isomorphism Conjecture. As a consequence, we show that the lower K-theory of the above group vanishes.