University of Cambridge > > Geometric Group Theory (GGT) Seminar > Generating RAAGs in 1-relator groups

Generating RAAGs in 1-relator groups

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  • UserAshot Minasyan, Southampton
  • ClockFriday 16 February 2024, 13:45-14:45
  • HouseMR13.

If you have a question about this talk, please contact Macarena Arenas.

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.

This talk is part of the Geometric Group Theory (GGT) Seminar series.

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