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Does every finitely presented subdirect product of limit groups admit a polynomial isoperimetric inequality? That is, does there exist a constant $C > 0$ and an integer $d \geq 1$ such that for every null-homotopic word $w$ in the generators of the group, there exists a van Kampen diagram for $w$ with at most $C|w|^d$ 2-cells, where $|w|$ denotes the length of the word $w$?

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  • $\begingroup$ Note: there is another natural slightly stronger property: to have all asymptotic cones simply connected. It is characterized without asymptotic cones as: for some $M$, every large enough loop can be split into $\le M$ loops of half its size. $\endgroup$
    – YCor
    Commented Aug 10 at 18:51

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In this 2024 preprint, your question is attributed as a conjecture of Bridson. See Conjecture 1.2, and note that a subdirect product of limit groups is the same thing as a residually free group. Therefore, it looks like this is an open question. It even appears to be open when all the limit groups are free.

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