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It is well known that all asymptotic cones simply connected implies polynomial Dehn function (Gromov). It is also well known that quadratic Dehn function implies all asymptotic cones simply connected (Papasoglu). Sapir and Olshanskii shows that there exists groups with cubic Dehn function, linear isodiametric function and no simply connected cones. Do we know how to construct a class of groups with arbitrary polynomial Dehn function and simply connected cones?

Added (from comment): Do we have a clue about a sufficient condition on the relations that will ensure a finitely presented group to have simply connected cones ?

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    $\begingroup$ Yes: the free $k$-step nilpotent group on $d\ge 2$ generators for $k\ge 3$, which has Dehn function $\simeq n^{k+1}$ (easy and well-known) and simply connected asymptotic cones (Pansu). $\endgroup$
    – YCor
    Apr 2 '14 at 21:39
  • $\begingroup$ One reference for Dehn functions of free nilpotent groups of class $c$ equal to $n^c$ is: G. Baumslag, C.F. Miller III, H. Short, Isoperimetric inequalities and the homology of groups, Invent. Math. 113 (1993), 531–560. Steve Gersten had another proof (also in 1993). They proved lower bound. the upper bound was provide by Ch. Pittet, "Isoperimetric inequalities for homogeneous nilpotent groups", 1995. $\endgroup$
    – Misha
    Apr 2 '14 at 22:45
  • $\begingroup$ Thanks for both answers. Do we have a clue about a sufficient condition on the relations that will ensure a finitely presented group to have simply connected cones ? $\endgroup$
    – Zenon
    Apr 3 '14 at 7:02
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"Do we have a clue about a sufficient condition on the relations that will ensure a finitely presented group to have simply connected cones ?" I do not think we have any clue. You can look at the papers by Curtis Kent for the latest info about this. For example, Kent, Curtis Asymptotic cones of HNN extensions and amalgamated products. Algebr. Geom. Topol. 14 (2014), no. 1, 551–595.

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  • $\begingroup$ Of course we do, just assume that the presentation satisfies a small cancellation condition, ensuring either hyperbolicity or CAT(0). (The harder question would be about necessary conditions and, I think, this is where we indeed do not have a clue.) $\endgroup$ Dec 10 '20 at 20:16
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    $\begingroup$ @MoisheKohan: It is not known that small cancelation conditions (say, C4-T4), imply CAT(0). It is an interesting open problem. Small cancelation does imply quadratic Dehn function, so Papasoglu's result applies. But this is a very small subset of the set of groups with simply connected asymptotic cones. And except for virtually nilpotent groups we know nothing about the set. Kent's papers, as far as I know, contain necessary and sufficient conditions but these are hard to check. $\endgroup$
    – markvs
    Dec 10 '20 at 20:31
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A f.g. group is a finitely presented group if and only if every large enough loop in the Cayley graph can be filled by a finite number of loops of length $\le$ half its length.

A f.g. group has all its asymptotic cones simply connected if and only if every large enough loop in the Cayley graph can be filled by a bounded (finite) number of length $\le$ half its length.

This is a full (asymptotic cone)-free characterization. But of course it doesn't mean it's in practice easy to determine. For instance, even for polycyclic group the picture is not yet clear which ones satisfy this property.

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