A week ago there was a preprint posted on the arXiv by Peter Haïssinsky and Cyril Lecuire about *Quasi-isometric rigidity of three manifold groups* (arXiv:2005.06813). Building on work by many other people, they complete the proof that the class of 3-manifold groups is quasi-isometrically rigid, meaning the following: if a finitely generated group G is quasi-isometric to a fundamental group of a compact 3-manifold, then G has a finite index subgroup which is in fact the fundamental group of a compact 3-manifold.

3-manifold are a special class of groups, because not every (finitely presented) group can be the fundamental group of a compact 3-manifold, contrary to the higher-dimensional case. An extended survey about (combinatorial) group theory of 3-manifold groups may be found in arXiv:1205.0202 and there is also a short and informative blog entry about the isomorphisms problem for 3-manifold groups at the Low Dimensional Topology blog.

The new arXiv preprint mentioned at the beginning of this post raised my interest in the large scale properties of 3-manifold groups. I compiled the following very short list of facts that I find interesting in this regard:

- John Mackay and Alessandro Sisto proved that all 3-manifold groups have finite asymptotic dimension (arXiv:1207.3008). This implies several things, for example that they satisfy all coarse isomorphism conjectures (arXiv:1607.03657) and that they are stably hypereuclidean (https://link.springer.com/article/10.1007/s10711-005-9025-0).
- Alessandro Sisto proved that all the asymptotic cones of a fixed 3-manifold group are bilipschitz equivalent to each other (arXiv:1109.4674).
- Martin Bridson proved that every 3-manifold group is asynchronously combable (https://link.springer.com/article/10.1007/BF01895689).

Let me end this post with a question to which I could not immediately find an answer for by just searching around the internet a bit: *Does every (infinite) 3-manifold group have a nice boundary-at-infinity?* Such boundaries can be easily constructed provided the group admits a suitable combing (arXiv:1711.06836), but unfortunately, the combings that Martin Bridson constructed (mentioned above in the last bullet point) are not (yet known to be) suitable in this regard.

Let me maybe also specify what I mean by a “nice” boundary-at-infinity: One way to define this would be to demand that if we put the boundary onto the universal cover of the corresponding compact 3-manifold, then it becomes an EZ-structure for that manifold (see e.g. the paper Local Homology Properties of Boundaries of Groups by Bestvina for this notion).