Partial word orders on groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:20:00Z http://mathoverflow.net/feeds/question/42981 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42981/partial-word-orders-on-groups Partial word orders on groups Mark Sapir 2010-10-21T01:40:32Z 2010-11-01T16:10:08Z <p>This is a followup question related to <a href="http://mathoverflow.net/questions/42928/orders-and-length-functions-on-finitely-generated-groups" rel="nofollow">this </a> question. Recall that a left-invariant partial order on a finitely generated group \$G\$ is called a partial <em>word order</em> 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. </p> <p><b> Question 1. </b> Is there a canonical way to construct non-trivial partial word orders on groups? </p> <p><b> Update 2</b> 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.)</p> <p><b> Update 1</b> <em>One possible generalization.</em> I call a partial order on a group <em>quasi-invariant</em> 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\$. </p> <p><b> Question 2. </b> Did anybody study quasi-invariant partial orders on groups?</p> <p><b> Motivation</b> 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. </p> http://mathoverflow.net/questions/42981/partial-word-orders-on-groups/44424#44424 Answer by grok for Partial word orders on groups grok 2010-11-01T07:47:49Z 2010-11-01T07:47:49Z <p>If G is a torsion group, then no elements may be comparable, so the only left-invariant order is the trivial one. That seems to rule out a canonical construction.</p>