0
$\begingroup$

Suppose we know the following about a class of groups $\mathcal{G}$.

  1. If $G$, $H$ are f.g. r.p. nonamenable groups, then $G \times H \in \mathcal{G}$.
  2. If $G \in \mathcal{G}$, $G$ is f.p., and $G$ is quasi-isometric to $H$, then $H \in \mathcal{G}$.

Here, f.g. means finitely generated, f.p. means finitely presented, r.p. means recursively presented.

What interesting/surprising/non-trivial corollaries can we deduce?

The motivation for this question is that I have a class for which I believe I can prove 1 and 2, so I'd like to know what interesting things my class contains just based on that. I already know a very interesting corollary, namely that my class contains a simple group Burger-Mozes. Unbelievable! Are there other surprises?

If it simplifies things, don't get too hung up on the r.p. and f.p. details. For instance if you can say something about the title, i.e. QI-closure of $\mathrm{NA}\times\mathrm{NA}$.

Improve this question
$\endgroup$
13
  • $\begingroup$ There appear to be at least three (1 2 3) "Burger–Mozes"s. Which one is the relevant one here? $\endgroup$
    – LSpice
    Commented May 19, 2020 at 20:49
  • $\begingroup$ Axiom 2 sounds weirdly stated, since $H$ QI to $G$ fp implies $G$ fp. So you have an isomorphism-closed class $\mathcal{G}$ of groups such that its subclass of fp elements is QI-closed, and contains all direct products of 2 fg rp non-amenable groups. The smallest class satisfying thus consists of (a) the fp groups QI to such a fp product (b) the rp fg direct products of two non-amenable groups. Probably the last formulation of the question is better $\endgroup$
    – YCor
    Commented May 19, 2020 at 20:50
  • $\begingroup$ @LSpice in this case it probably means simple f.p. groups that occur as acting geometrically on a product of two trees. Other interesting groups QI to a product of non-amenable groups (and not virtually directly decomposable) are various irreducible lattices in products of Lie or non-archimedean groups. Rataggi also produced groups QI to a product of two bushy trees, that have no proper finite index subgroups, but have many normal subgroups of infinite index. $\endgroup$
    – YCor
    Commented May 19, 2020 at 20:51
  • $\begingroup$ BTW "interesting" obviously depends on how your class is defined... I somewhat have the guess that your class is that of groups that admit a strongly aperiodic subshift, or something of this spirit, but this is hard to guess a priori... $\endgroup$
    – YCor
    Commented May 19, 2020 at 22:46
  • $\begingroup$ I added the correct BM reference, the deduction is on page 760 of Drutu-Kapovich. $\endgroup$
    – Ville Salo
    Commented May 19, 2020 at 23:34

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .