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this is my first question so let me know if I broke any rules. I am writing my dissertation on the zero divisor conjecture in group rings, and I am struggling to find any examples I could put in my chapter examining the classes of groups mentioned above. This might be a dumb question, but I have searched for hours on google scholar and through the books at my university library, but I never find anything like

$$G = \langle \text{generating elements} \mid \text{relation of elements} \rangle.$$

Ideally, I'm looking for a group that satisfies the unique product condition, but is not orderable, and an orderable group that is not free. If anyone knows any literature that could help me, that would be greatly appreciated too!

Arthur
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Gregory Alexander Swart
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  1. There are many examples of non-free right-orderable groups, with explicit (finite) presentations for instance, surface groups, Braid groups, torsion-free 1-relator groups $$ \langle x_1,...,x_n| w\rangle, $$ i.e. groups where $w$ is a cyclically reduced word which is not a proper power of another word.

  2. Examples of groups with the UP property which are not right-orderable (RO) are harder to find, one such example is given by Dunfield in his appendix to

S. Kionke, J. Raimbault, On geometric aspects of diffuse groups. With an appendix by Nathan Dunfield. Doc. Math. 21 (2016), 873–915.

Specifically, he gives an example (with an explicit presentation) of a diffuse group which is not RO. On the other hand, diffuse groups are proven to have the UP property in

B. Bowditch, A variation on the unique product property. J. London Math. Soc. (2) 62 (2000), no. 3, 813–826.

  1. Just for completeness, here is the definition of the UP (Unique Product) Property:

A group $G$ is said to have the unique product property if for every two finite non-empty subsets $A, B \subset G$ there is an element in the product $x \in A \cdot B$ for which the representation in the form $x = ab$ with $a \in A$ and $b \in B$ is unique. (I.e., if $x=a_1b_1=a_2b_2, a_i\in A, b_i\in B, i=1,2$, then $a_1=a_2, b_1=b_2$.)

The interest in such groups comes from the (rather easy) fact that very group satisfying the UP property has no zero divisors in the group ring ${\mathbb Z}G$. Such $G$ is necessarily torsion-free. For a while, it was unknown if every torsion-free group has the UP property. The first examples were constructed in

E. Rips and Y. Segev, Torsion-free group without unique product property, J. Algebra, 108 (1987), no. 1, 116–126.

Moishe Kohan
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