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This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have $|b|\le C(|a|+|c|)$ for some constant $C$ (where $|x|$ is the word length of $x$). For example, the following partial order on ${\mathbb Z}^2$: $(m,n)\le (k,l)$ iff $m\le k, n\le l$ is a word partial order. Certainly the empty (trivial) partial order ($a\le b$ iff $a=b$) is always a word order.

Question 1. Is there a canonical way to construct non-trivial partial word orders on groups?

Update 2 A sub-question: is there a general algebraic property of a group that guaranties existence of such non-trivial partial orders? (Being free Abelian of finite rank is such a property, but I am looking for non-trivial answers.)

Update 1 One possible generalization. I call a partial order on a group quasi-invariant if for every $g,a,b$ if $a\lt b$, then there exist two elements $c,d$ such that $gac\lt gbd$ and $|c|,|d|\lt C$ for some constant $C$.

Question 2. Did anybody study quasi-invariant partial orders on groups?

Motivation The reason I want to study such things is to introduce an extra structure on the asymptotic cones of groups. If one tries to define a partial order on an asymptotic cone, one would need a quasi-(left)-invariant word partial order on the group.

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This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have $|b|\le C(|a|+|c|)$ for some constant $C$ (where $|x|$ is the word length of $x$). For example, the following partial order on ${\mathbb Z}^2$: $(m,n)\le (k,l)$ iff $m\le k, n\le l$ is a word partial order. Certainly the empty (trivial) partial order ($a\le b$ iff $a=b$) is always a word order.

Question 1. Is there a canonical way to construct non-trivial partial word orders on groups?

Update One possible generalization. I call a partial order on a group {\em quasi-invariant} if for every $g,a,b$ if $a\lt b$, then there exist two elements $c,d$ such that $gac\lt gbd$ and $|c|,|d|\lt C$ for some constant $C$.

Question 2. Did anybody study quasi-invariant partial orders on groups?

Motivation The reason I want to study such things is to introduce an extra structure on the asymptotic cones of groups. If one tries to define a partial order on an asymptotic cone, one would need a quasi-(left)-invariant word partial order on the group.

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