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.

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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 .

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