Abstract

Given a finite simplicial graph $\Gamma$, the right angled Artin group (RAAG) $A(\Gamma)$ is generated by the vertices of $\Gamma$ subject to the relations that two vertices commute if and only if they are adjacent in $\Gamma$. The monoid with the same presentation is called the trace monoid $T(\Gamma)$. RAA Gs play an important role in Geometric Group Theory, while Trace monoids originated in Computer Science.

The trace monoid $T(\Gamma)$ is naturally embedded in the RAAG $A(\Gamma)$, as the set of positive words. In my talk I will discuss the following problem: suppose that a 1-relator group G contains a submonoid isomorphic to $T(\Gamma)$. Does $G$ also contain a copy of $A(\Gamma)$ as a subgroup?

This problem is motivated by recent work of Foniqi, Gray and Nyberg-Brodda, who proved that groups containing T(P_4), where P_4 is the path with 4 vertices (of length 3), have undecidable rational subset problem. They also exhibited 1-relator groups containing A(P_4) and asked whether every 1-relator group which has a submonoid isomorphic to T(P_4) must also have a subgroup isomorphic to A(P_4). I will sketch an argument, based on joint work with Motiejus Valiunas (University of Wroclaw, Poland), showing that the answer to the latter question is positive.