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http -> https (the question has been bumped anyway)

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edited Jan 30, 2020 at 20:51

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There are also lots of amenable examples: by a theorem of P. Linnell and D. Witte-Morris (in this paper this paper), an amenable group is left-orderable if and only if it is locally indicable (i.e. any non-trivial subgroup surjects onto $\mathbb Z$). It is not hard to see that this implies that any crystallographic group with non-solvable holonomy is not left-orderable; on the other hand any such group contains a free abelian group of finite index, and the latter is certainly orderable. One can also get examples more complicated olycyclic examples.

A simpler crystallographic example is given by the subgroup of ${\rm Isom}(\mathbb{R}^3)$ generated by three half-turns with disjoint axes of orthogonal directions (its abelianization is finite; in fact it has a presentation given by $$ \langle a,b | ab^2a^{-1}=b^{-2},\, ba^2b^{-1} = a^{-2}\rangle $$ (see for example the last section in this paper this paper of Bowditch).

There are also lots of amenable examples: by a theorem of P. Linnell and D. Witte-Morris (in this paper), an amenable group is left-orderable if and only if it is locally indicable (i.e. any non-trivial subgroup surjects onto $\mathbb Z$). It is not hard to see that this implies that any crystallographic group with non-solvable holonomy is not left-orderable; on the other hand any such group contains a free abelian group of finite index, and the latter is certainly orderable. One can also get examples more complicated olycyclic examples.

A simpler crystallographic example is given by the subgroup of ${\rm Isom}(\mathbb{R}^3)$ generated by three half-turns with disjoint axes of orthogonal directions (its abelianization is finite; in fact it has a presentation given by $$ \langle a,b | ab^2a^{-1}=b^{-2},\, ba^2b^{-1} = a^{-2}\rangle $$ (see for example the last section in this paper of Bowditch).

There are also lots of amenable examples: by a theorem of P. Linnell and D. Witte-Morris (in this paper), an amenable group is left-orderable if and only if it is locally indicable (i.e. any non-trivial subgroup surjects onto $\mathbb Z$). It is not hard to see that this implies that any crystallographic group with non-solvable holonomy is not left-orderable; on the other hand any such group contains a free abelian group of finite index, and the latter is certainly orderable. One can also get examples more complicated olycyclic examples.

A simpler crystallographic example is given by the subgroup of ${\rm Isom}(\mathbb{R}^3)$ generated by three half-turns with disjoint axes of orthogonal directions (its abelianization is finite; in fact it has a presentation given by $$ \langle a,b | ab^2a^{-1}=b^{-2},\, ba^2b^{-1} = a^{-2}\rangle $$ (see for example the last section in this paper of Bowditch).

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created Sep 8, 2014 at 11:18

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There are also lots of amenable examples: by a theorem of P. Linnell and D. Witte-Morris (in this paper), an amenable group is left-orderable if and only if it is locally indicable (i.e. any non-trivial subgroup surjects onto $\mathbb Z$). It is not hard to see that this implies that any crystallographic group with non-solvable holonomy is not left-orderable; on the other hand any such group contains a free abelian group of finite index, and the latter is certainly orderable. One can also get examples more complicated olycyclic examples.

A simpler crystallographic example is given by the subgroup of ${\rm Isom}(\mathbb{R}^3)$ generated by three half-turns with disjoint axes of orthogonal directions (its abelianization is finite; in fact it has a presentation given by $$ \langle a,b | ab^2a^{-1}=b^{-2},\, ba^2b^{-1} = a^{-2}\rangle $$ (see for example the last section in this paper of Bowditch).