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> (1) I am looking for an example of a u.p (unique product) group which is not right orderable (RO). 
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Almost any group I pick up (obviously torsion-free, as u.p. group cannot have nontrivial torsion elements) turns out to be RO, whether it be easy groups like integers, modulo $n$ etc, or groups on matrices, or free groups, etc.  

An example of a torsion-free non-RO group can be found [at this question](https://mathoverflow.net/a/81106/14094).


A group $G$ is said to be u.p., if given any two finite nonempty subsets $A$ and $B$ of $G$, there exist at least one element $x$ which has a unique representation $x=ab$ where $a \in A$ and $b \in B$.




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Similarly, a two unique product group (t.u.p. group) is a group $G$ such that if given any two finite nonempty subsets $A$ and $B$ of $G$ with $|A|+|B| > 2$, there exist at least two elements $x$ and $y$ which have unique representations $x=ab$ and $y=cd$ where $a,c \in A$ and $b,d \in B$.



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> (2) Is there a u.p. group which is not t.u.p.? (clearly, every t.u.p. group is a u.p. group).
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As every RO-group is a t.u.p. group, which in turn is u.p. group, (1) was a natural question that came to my mind.



Unique product group which is not right orderable
Such a group has been found by N. Dunfield, see the appendix to
the paper


* Steffen Kionke, Jean Raimbault, Nathan Dunfield, *On geometric aspects of diffuse groups*, Documenta Mathematica, Vol. 21 (2016), 873-915, [journal](https://www.math.uni-bielefeld.de/documenta/vol-21/24.html), arXiv:[1411.6449](https://arxiv.org/abs/1411.6449).


The group is the fundamental group of a compact hyperbolic three--manifold which has injectivity radius large enough so that it is known to have unique products (and a little more) by a result of Delzant--Bowditch, but Nathan checked "by hand" that it is not left-orderable (by the same method as in his Inventiones paper with D. Calegari,


* Danny Calegari, Nathan M. Dunfield, *Laminations and groups of homeomorphisms of the circle*, Invent. Math. **152** (2003) 149-207, doi:[10.1007/s00222-002-0271-6](https://doi.org/10.1007/s00222-002-0271-6), arXiv:[math/0203192](https://arxiv.org/abs/math/0203192)


which you should check out if you want more examples of non-left/right-orderable groups).


