Arranged group

In group theory , a sub- discipline of mathematics , an arranged group ( left-orderable group ) is a group together with a total order " " that is compatible with the left translation given by the multiplication. Well-known examples are the groups of whole and real numbers . ${\displaystyle <}$

definition

Be a group. A left-invariant arrangement on is a total order, so that applies to all : ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle a,b,c\in G}$

${\displaystyle a<b\Longleftrightarrow ca<cb}$.

An ordered group is a group with a left-invariant order.

Equivalently, a left-invariant order can be characterized by a disjoint decomposition

${\displaystyle G=P\cup N\cup \left\{Id\right\}}$

with and . ${\displaystyle P\cdot P\subset P}$ ${\displaystyle P^{-1}=N}$

The arrangement results from the decomposition via

${\displaystyle a<b\Longleftrightarrow a^{-1}b\in P}$.

Examples

• ${\displaystyle \mathbb {Z} }$and are arranged groups. ${\displaystyle \mathbb {R} }$
• If there are torsional (i.e., finite order ) elements in a group , then the group cannot have a left-invariant ordering.
• Each torsion-free Abelian group is an ordered group.
• Free groups are arranged.
• SL (2, R) and have no left-invariant arrangement. ${\displaystyle SL(2,\mathbb {Z} \left[{\frac {1}{p}}\right])}$
• Dehornoy's theorem : groups of braids are arranged.
• Rourke-Wiest theorem : mapping class groups of surfaces with non-empty edges are arranged.
• Boyer-Rolfsen-Wiest theorem : Fundamental groups of compact, irreducible 3-manifolds with are arranged. ${\displaystyle \pi _{1}M}$ ${\displaystyle P^{2}}$ ${\displaystyle M}$ ${\displaystyle b_{1}(M)>0}$
• If and are arranged groups and ${\displaystyle K}$ ${\displaystyle H}$

${\displaystyle 0\rightarrow K\rightarrow G\rightarrow H\rightarrow 0}$

is a short exact sequence then has a left-invariant ordering compatible with that of and for which the mapping is monotonic. ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle G\rightarrow H}$

• Burns-Hale theorem : A group has a left-invariant arrangement if there is a surjective homomorphism on an arranged group for every finitely generated subgroup . In particular, has a left-invariant arrangement when for each finitely generated subgroup applies: . ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle \phi :H\rightarrow L_{H}}$ ${\displaystyle L_{H}\not =1}$ ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle H^{1}(H,\mathbb {Z} )\not =0}$
• The universal overlay of is an ordered group, although all of your finitely generated subgroups apply. ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle H^{1}(H,\mathbb {Z} )=0}$ ${\displaystyle H}$
• A countable group has a left-invariant assembly if and only if a subset of isomorphic , the group of the orientation -prolonging homeomorphisms of is. ${\displaystyle Homeo^{+}(\mathbb {R} )}$ ${\displaystyle \mathbb {R} ^{1}}$
• Hölder's theorem : A group has a left-invariant Archimedean arrangement if and only if it is isomorphic to a subgroup of . ${\displaystyle \mathbb {R} }$

Bi-invariant arrangements

A right-invariant arrangement on a group is a total order, so that applies to all : ${\displaystyle G}$ ${\displaystyle a,b,c\in G}$

${\displaystyle a<b\Longleftrightarrow ac<bc}$.

Each arranged group also has a right-invariant arrangement, but this generally does not correspond to the left-invariant arrangement.

A bi-invariant arrangement is an arrangement that is left and right invariant at the same time. For example, torsion-free Abelian groups or the pure braid group have a bi-invariant arrangement.

See also

• Ordered Abelian group

literature

• Robert G. Burns, VWD Hale: A note on group rings of certain torsion-free groups. In: Canadian Mathematical Bulletin. Vol. 15, No. 3, 1972, pp. 441-445, doi : 10.4153 / CMB-1972-080-3 .
• Danny Calegari : Circular groups, planar groups, and the Euler class. In: Cameron Gordon , Yoav Rieck (eds.): Proceedings of the Casson Fest (Arkansas and Texas 2003) (= Geometry & Topology Monographs. Vol. 7, ISSN  1464-8989 ). University of Warwick - Mathematics Institute, Coventry 2004, pp. 431-491, doi : 10.2140 / gtm.2004.7.431 .
• Patrick Dehornoy , Ivan Dynnikov, Dale Rolfsen, Bert Wiest: Why are braids orderable? (= Panoramas et Synthèses. Vol. 14). Société Mathématique de France, Paris 2002, ISBN 2-85629-135-X .

Web links

• Clay, Rolfsen: Ordered groups and topology
• Deroin, Neves, Rivas: Groups, Orders and Dynamics