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Is it true that the Baumslag-Solitar groups, say, $BS(1,n)$, $|n|\ge 2$, are finitely presented groups with largest Dehn functions (namely, exponential growth) known to be inside finitely presented groups with quadratic Dehn functions?

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  • $\begingroup$ By the way, (1) what do you know about possible Dehn functions of f.p. subgroups of hyperbolic groups? (this is part of your question since if $H$ is infinite hyperbolic, then $H\times\mathbf{Z}$ has quadratic Dehn function). Is the Dehn function always polynomially bounded ?(2) What about finitely presented subdirect products of free groups? again, is the Dehn function always polynomially bounded? (3) same with subgroups of CAT(0) groups. $\endgroup$
    – YCor
    Sep 18, 2018 at 9:19
  • $\begingroup$ Another remark. If we take Thompson's group $F$ (which has a quadratic Dehn function, a result of Guba) acting on $[0,1]$ and restrict to its subgroup $L$ of elements $g$ with $u_0(g)u_1(g)=1$, where $u_i$ is the slope at $i$, then $L$ is f.p., as an ascending HNN extension of copies of $F$. I suspect that $L$ also has an exponential Dehn function. $\endgroup$
    – YCor
    Sep 18, 2018 at 9:23
  • $\begingroup$ @YCor: I do not think you will find any f.p. subgroup of $F$ with more than polynomial Dehn function. Of course that would be a great result but that is where our research with Gili Golan is heading now. A more promising would be linear groups but, in a strange way, there are a lot of similarities between $F$ and $SL_n(\mathbb{Z})$. $F$ also has closed subgroups which have low Dehn functions, etc. So if you do not know other examples, your result with Romain is the current record? $\endgroup$
    – user6976
    Sep 18, 2018 at 11:47
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    $\begingroup$ We just proved with Olshanskii that groups with quadratic Dehn functions contain groups with arbitrary large recursive Dehn function. That was the reason I asked my question. But that is as far as we can go. The answer to your question is unknown. I suspect that the answer is "no". $\endgroup$
    – user6976
    Sep 19, 2018 at 16:50
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    $\begingroup$ @Ycor: Sorry! The answer to your question is of course "no". The reason is that if a f.g. group $G$ is inside a group with quadratic Dehn function $H$, then the word problem in $G$ can be solved (non-deterministically) in quadratic time, and determministiclly in $\exp$ of that. So if you take a f.p. group with very complicated word problem, it cannot embed into a group with quadratic Dehn function. $\endgroup$
    – user6976
    Sep 19, 2018 at 17:45

2 Answers 2

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Yes, the highest known Dehn function of a subgroup of a finitely presented group with quadratic Dehn function is exponential. The example was found by Yves Cornulier and Romain Tessera in Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups. Confluentes Math. 2 (2010), no. 4, 431–443.

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As mentioned in the comments, Sapir and Olshanskii recently proved in Algorithmic problems in groups with quadratic Dehn functions that:

Theorem. For every recursive function $f$, there exist finitely presented groups $H \leq G$ such that $G$ has quadratic Dehn function and such that $H$ has Dehn function at least $f$.

However, I would like to mention a construction that predates Cornulier and Tessera's article cited in the previous answer: In their article Finitely presented subgroups of automatic groups and their isoperimetric functions, Baumslag, Bridson, Miller III, and Short proved that:

Theorem. There exist a biautomatic group $B$ and a finitely presented subgroup $G \leq B$ such that $G$ is not of type $FP_3$ and its isoperimetric function is strictly exponential. Moreover one can arrange for $B$ to be the fundamental group of a closed manifold of non-positive curvature.

In this case, the overgroup has quadratic Dehn function but it can also be chosen to be CAT(0). In this direction, I probably should mention the very recent preprint Superexponential Dehn functions inside CAT(0) groups in which finitely presented subgroups with superexponential Dehn functions are constructed in CAT(0) groups.

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  • $\begingroup$ Just to make it clear, the paper [CT] (arXiv link) mentioned by Mark Sapir doesn't claim to yield the first embedding of a group with exponential Dehn function into one with quadratic Dehn function, but claims it for embeddings of solvable Baumslag-Solitar groups. (And the larger group turns out to also be metabelian, of finite Prüfer rank). A side question asked in [CT] (line 15 of introduction) is whether every f.g. group with solvable word problem can be embedded into a f.p. group with quadratic Dehn function. $\endgroup$
    – YCor
    Mar 1, 2021 at 9:44

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