Borel tightness kit

The tightness set of Borel (engl .: Borel density theorem ) is a theorem of the mathematics , the grating in algebraic groups, such as, for example, in characterized. ${\displaystyle SL(n,\mathbb {Z} )}$ ${\displaystyle SL(n,\mathbb {R} )}$

It says that every polynomial function that vanishes on a lattice must be identical to 0 on the entire algebraic group . ${\displaystyle \Gamma \subset G}$ ${\displaystyle G}$

sentence

Let be a connected semisimple - algebraic group with no compact factor and be a lattice in . ${\displaystyle G}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle G}$

Then Zariski - close in . ${\displaystyle \Gamma }$ ${\displaystyle G}$

Applications

In the following we assume that and meet the requirements of the leak tightness kit. ${\displaystyle G}$ ${\displaystyle \Gamma }$

• If is an irreducible polynomial representation of , then the restriction of to is also an irreducible representation. ${\displaystyle \phi \colon G\to GL(m,\mathbb {R} )}$ ${\displaystyle G}$ ${\displaystyle \phi }$ ${\displaystyle \Gamma }$
• If a connected, closed subgroup of is normalized , then it is a normal subgroup of . ${\displaystyle H\subset G}$ ${\displaystyle \Gamma }$ ${\displaystyle G}$
• The centralizer of in is the center of . ${\displaystyle \Gamma }$ ${\displaystyle G}$ ${\displaystyle Z(G)}$ ${\displaystyle G}$
• Every finite normal divisor of is contained in. ${\displaystyle \Gamma }$ ${\displaystyle Z(G)}$
• ${\displaystyle \Gamma }$is a subset of finite index in its normalizer.
• There is a decomposition so that an irreducible lattice is commensurable in and with . ${\displaystyle G=G_{1}\times \ldots \times G_{r}}$ ${\displaystyle \Gamma _{i}=\Gamma \cap G_{i}}$ ${\displaystyle G_{i}}$ ${\displaystyle \Gamma _{1}\times \ldots \times \Gamma _{r}}$ ${\displaystyle \Gamma }$
• For polynomial functions on the following applies: ${\displaystyle Q\in \mathbb {R} \left[x_{11},\ldots ,x_{nn}\right]}$ ${\displaystyle Mat(n\times ,n,\mathbb {R} )}$

${\displaystyle Q(\Gamma )=0\Longrightarrow Q(G)=0.}$

literature

• Armand Borel : Density properties for certain subgroups of semi-simple groups without compact components . Ann. of Math. (2) 72, 179-188, 1960.
• MS Raghunathan: Discrete subgroups of Lie groups . Results of mathematics and its border areas. Volume 68.Berlin-Heidelberg-New York: Springer-Verlag, 1972.
• RJ Zimmer: Ergodic theory and semisimple groups . Monographs in Mathematics, Vol. 81.Boston-Basel-Stuttgart: Birkhäuser, 1984.
• D. Witte Morris: Introduction to arithmetic groups . Deductive Press, 2015. ISBN 978-0-9865716-0-2 / pbk 978-0-9865716-1-9 / hbd