Trace monoids and RAAGs in one-relator groups

A right-angled Artin group (RAAG; resp. a trace monoid) is a finitely presented group (resp. monoid) in which all relations are those making some pairs of the generators commute. Embeddings of trace monoids in one-relator groups have been recently studied in order to construct one-relator groups with undecidable submonoid and prefix membership problems. In an attempt to describe one-relator groups with undecidable rational subset membership problem, the following question arose: does there exist a one-relator group containing the trace monoid whose defining graph is the path of length 3, but not the corresponding RAAG? In this talk, I will use splittings of one-relator groups over free subgroups to explain why the answer is "no", even if the path of length 3 is replaced by any finite tree. This is joint work with Ashot Minasyan.