Is it true that the BaumslagSolitar 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?
13

$\begingroup$ 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 '18 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 '18 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? $\endgroup$ – Mark Sapir Sep 18 '18 at 11:47

1$\begingroup$ 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". $\endgroup$ – Mark Sapir Sep 19 '18 at 16:50

2$\begingroup$ @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 (nondeterministically) 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. $\endgroup$ – Mark Sapir Sep 19 '18 at 17:45
5
$\begingroup$
$\endgroup$
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 BaumslagSolitar groups. Confluentes Math. 2 (2010), no. 4, 431–443.