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In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2).

The symmetries are called dilatations, and they are the maps on points that send any line to a parallel line. A translation is defined to be a dilatation that fixes no point (or is the identity map). It is then proved that translation has a "direction" (a partition of the point set into parallel lines that are sent to themselves by the translation).

It is then proved that (Theorem 2.8) if translations with different directions exist, then the group T of all translations is abelian. The following remark appears at the end of the section.

"REMARK. It is conceivable that the geometry contains only translations with a single direction (aside from the identity). It is an undecided question whether T has to be commutative in this case. Probably there do exist counterexamples, but none is known."

I'm working through this book with a student, but unfortunately I am not an expert on this material. The student and myself would like to know the current status of this Remark (the book was written in 1957). Can anyone provide such an example of a geometry with a nonabelian group of translations? Or point me to a reference?

Thanks.

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  • $\begingroup$ It would help if you reminded us of Artin's definition of a geometry. The Bass--Serre tree of the (1,2)-Baumslag--Solitar group has the kind of behaviour you ask for, but I don't know if it qualifies. $\endgroup$ – HJRW Sep 24 '16 at 6:11
  • $\begingroup$ Artin's geometry consists of a set of points and a set of lines. A pair of points determines exactly one line. For each line and point, there is a unique line through the point parallel to the given line. The only other axioms assert the existence of enough points and symmetries. I'll have a look at your suggestion, thanks. $\endgroup$ – Josh Sep 24 '16 at 17:16
  • $\begingroup$ In that case, the parallel postulate probably rules out these kinds of examples. $\endgroup$ – HJRW Sep 25 '16 at 13:11

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