Convergence of Baumslag-Solitar groups

Abstract

We study convergent sequences of Baumslag-Solitar groups in the space of marked groups. We prove that $BS(\mathfrak m,\mathfrak n) \to \F_2$ for $|\mathfrak m|,|\mathfrak n| \to \infty$ and $BS(1,\mathfrak n) \to \mathbb{Z}\wr\mathbb{Z}$ for $|\mathfrak n| \to \infty$. For $\mathfrak m$ fixed, $|\mathfrak m| \geqslant 2$, we show that the sequence $(BS(\mathfrak m,\mathfrak n))_{\mathfrak n}$ is not convergent and characterize many convergent subsequences. Moreover if $X_\mathfrak m$ is the set of $BS(\mathfrak m,\mathfrak n)$'s for $\mathfrak n$ relatively prime to $\mathfrak m$ and $|\mathfrak n| \geqslant 2$, then the map $BS(\mathfrak m,\mathfrak n) \mapsto \mathfrak n$ extends continuously on $\overline{X_\mathfrak m}$ to a surjection onto invertible $\mathfrak m$-adic integers.