Dimensional equivalence

In the mathematical field of group theory , the measure equivalence of groups is a weakening of the concept of quasi-isometry.

In contrast to the geometric group theory are classified in groups of up to quasi-isometric, is the study of groups of up to Maßäquivalenz as a measurable group theory (ger .: measurable group theory ).

The Maßäquivalenz should not the equivalence of measures to be confused.

definition

Two countable groups and are called measure equivalent if there are commuting measure-maintaining free effects from and on a measure space with , both of which have a fundamental domain of finite measure. ${\displaystyle \Gamma }$ ${\displaystyle \Lambda }$ ${\displaystyle \Gamma }$ ${\displaystyle \Lambda }$ ${\displaystyle (\Omega ,m)}$ ${\displaystyle m(\Omega )=\infty }$

The space with the effects of and is called a coupling of and . The ratio of the volumes of the fundamental areas is called the coupling constant ${\displaystyle (\Omega ,m)}$ ${\displaystyle \Gamma }$ ${\displaystyle \Lambda }$ ${\displaystyle \Gamma }$ ${\displaystyle \Lambda }$

${\displaystyle c_{\Gamma ,\Lambda }:={\frac {vol(\Omega /\Lambda )}{vol(\Omega /\Gamma )}}}$.

Motivation of the definition

Gromov has shown that two finitely generated groups are quasi-isometric if and only if there are commuting, continuous effects on a locally compact space that are actually discontinuous and co-compact .

The definition of the equivalence of measure is in this sense a weakening of the definition of the quasi-isometry. However, it is an open question whether quasi-isometric groups are always dimensionally equivalent.

Examples

• Two grids and in a locally compact group are dimensionally equivalent. The group with the Hair measure as well as the links effect of and the legal effect of a coupling. (In contrast, these grids are generally not quasi-isometric; this only applies to co-compact grids .) ${\displaystyle \Gamma }$ ${\displaystyle \Lambda }$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \Gamma }$ ${\displaystyle \Lambda }$
• A countable group is dimensionally equivalent to the group of integers if and only if it is indirect . ${\displaystyle \mathbb {Z} }$
• Lattices in connected , simple Lie groups of rank with a trivial center are only dimensionally equivalent if is. ${\displaystyle \Gamma _{1}\subset G_{1},\Gamma _{2}\subset G_{2}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle rk_{\mathbb {R} }(G_{i})\geq 2}$ ${\displaystyle G_{1}=G_{2}}$

properties

Measure equivalence is an equivalence relation .

A measure equivalence is called ergodic if the effect of is on ergodic . Every measure equivalence has an ergodic decomposition as an integral of ergodic measure equivalences. ${\displaystyle \Gamma \times \Lambda }$ ${\displaystyle \Omega }$

Two countable groups are measure-equivalent if and only if they allow stable orbit-equivalent substantially free measure-maintaining effects .

Invariants

Let and be equivalent groups with coupling constant . Then: ${\displaystyle \Gamma }$ ${\displaystyle \Lambda }$ ${\displaystyle c=c_{\Gamma ,\Lambda }}$

• ${\displaystyle cost(\Lambda )-1=c(cost(\Gamma )-1)}$for the costs of the groups,
• ${\displaystyle b_{i}^{(2)}(\Lambda )=cb_{i}^{(2)}(\Gamma )\ \forall i\geq 0}$for the L ² -Betti numbers ,
• the ergodic dimensions of and agree. ${\displaystyle \Gamma }$ ${\displaystyle \Lambda }$

Representation theory

If two groups are metrologically equivalent, then each (unitary) representation of one group induces a (unitary) representation of the other.

In particular, property T , indirectness, and the Haagerup property are well-defined properties modulo measure equivalence.

literature

• Alex Furman : Gromov's measure equivalence and rigidity of higher rank lattices , Annals of Mathematics 150 (1999), 1059-1081. online (PDF)
• Damien Gaboriau : Orbit equivalence and measured group theory. Proceedings of the International Congress of Mathematicians. Volume III, 1501-1527, Hindustan Book Agency, New Delhi, 2010.

Web links

• Furman: A survey on measured group theory