Convergence of Baumslag-Solitar groups

Yves Stalder (1)

1 UNINE - Institut de Mathématiques

We study convergent sequences of Baumslag-Solitar groups in the space of marked groups. We prove that $BS({m},{ n}) \to\Bbb F_2$ for $|{m}|,|{ n}|\to\infty$ and $BS(1,{ n})\to{\Bbb Z}\wr{\Bbb Z}$ for $|{ n}|\to\infty$. For $ m$ fixed, $| m|\geqslant2$, we show that the sequence $(BS( m, n))_{ n}$ is not convergent and characterize many convergent subsequences. Moreover if $X_{ m}$ is the set of $BS( m, n)$'s for $ n$ relatively prime to $ m$ and $| n|\geqslant2$, then the map $BS( m, n)\mapsto n$ extends continuously on $\overline{X_{ m}}$ to a surjection onto invertible $ m$-adic integers.

Mots clés: Baumslag-Solitar groups; space of marked groups

Domaines: Théorie des groupes [math.GR]

Yves Stalder. Convergence of Baumslag-Solitar groups. Bulletin of the Belgian Mathematical Society - Simon Stevin, 2006, 13, pp.221-233. ⟨10.36045/bbms/1148059458⟩. ⟨hal-00472505⟩