Naomi Andrew, "Automorphisms behaving badly"

Baumslag–Solitar groups are HNN extensions of the infinite cyclic group, whose isomorphism type is controlled by two integers giving the two embeddings. They have provided many counterexamples over the years: for example, they include groups which are not Hopfian and groups which are Hopfian but not residually finite. Later, Collins and Levin showed that there are Baumslag–Solitar groups that do not have finitely generated automorphism group.

Moving this construction to higher rank, one can study "Leary–Minasyan groups": these are HNN extensions of free abelian groups, with both inclusions finite index. They are also sources of counterexamples, such as groups which are CAT(0) but not biautomatic. We study their automorphism groups, and in particular characterise when they are finitely generated; this includes some finitely presented metabelian groups with automorphism groups that are not finitely generated. This is joint work with Sam Hughes and Motiejus Valiunas.