Commensurator

The commensurator of subgroups is a term from group theory , a branch of mathematics .

definition

The Kommensurator the subgroup of a group , designated , or if the group is evident from the context, whether or not , is ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle \operatorname {comm} _{G}(H)}$ ${\displaystyle G}$ ${\displaystyle \operatorname {comm} (H)}$

${\displaystyle \operatorname {comm} _{G}(H)=\left\{g\in G:gHg^{-1}\cap H{\mbox{ hat endlichen Index in }}H{\mbox{ und }}gHg^{-1}\right\}}$.

The commensurator is a subset of . ${\displaystyle G}$

Examples

• If there is an Abelian group, then is for each subgroup . ${\displaystyle G}$ ${\displaystyle \operatorname {comm} _{G}(H)=G}$ ${\displaystyle H\subset G}$
• More generally, if is a normal divisor, then is . ${\displaystyle H\subset G}$ ${\displaystyle \operatorname {comm} _{G}(H)=G}$
• The commensurator of a factor in the free product is . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} *\mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$
• More generally, the Kommensurator quasiconvex a subset of a hyperbolic group is . ${\displaystyle H\subset G}$ ${\displaystyle G}$ ${\displaystyle H}$
• The commensurator of in is . ${\displaystyle SL(n,\mathbb {Z} )}$ ${\displaystyle SL(n,\mathbb {R} )}$ ${\displaystyle SL(n,\mathbb {Q} )}$

Application: characterization of arithmetic grids

Theorem ( Margulis ) : An irreducible lattice in a semi-simple Lie group is arithmetic if and only if has an infinite index in its commensurator, i.e. if . ${\displaystyle \Gamma \subset G}$ ${\displaystyle G}$ ${\displaystyle \Gamma }$ ${\displaystyle [\operatorname {comm} _{G}(\Gamma ):\Gamma ]=\infty }$

literature

• Zimmer, Robert J .: Ergodic theory and semisimple groups. Monographs in Mathematics, 81st Birkhäuser Verlag, Basel, 1984. ISBN 3-7643-3184-4