We present sufficient conditions for HNN extensions to be inner amenable, respectively ICC, which give necessary and sufficient criteria among Baumslag-Solitar groups. We deduce that such a group, viewed as acting on its Bass-Serre tree, contains non trivial elements which fix unbounded subtrees.
We study convergent sequences of Baumslag-Solitar groups in the space of marked groups. We prove that BS(m,n) --> F_2 for |m|,|n| --> infty and BS(1,n) --> Z wr Z for |n| --> infty. For m fixed, |m|>1, we show that the sequence (BS(m,n))_n is not con
vergent and characterize many convergent subsequences. Moreover if X_m is the set of BS(m,n)s for n relatively prime to m and |n|>1, then the map BS(m,n) mapsto n extends continuously on the closure of X_m to a surjection onto invertible m-adic integers.
