A closer look at the non-Hopfianness of $BS(2,3)$

DOI10.36045/J.BBMS.200507arXiv2005.03396MaRDI QIDQ6340237

Tom Kaiser

Publication date: 7 May 2020

Abstract: The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $phi$ onto itself, that is not injective. In particular this is equivalent to saying that $BS(2,3)$ has a non-trivial quotient that is isomorphic to itself. As a consequence the Cayley graph of $BS(2,3)$ has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism $phi$. We show its kernel is a free group of infinite rank with an explicit set of generators. Finally we show how $phi$ appears as a morphism on fundamental groups induced by some continuous map. This point of view was communicated to the author by Gilbert Levitt.

Mathematics Subject Classification ID

Groups acting on trees (20E08)

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