Theorem of Borel and Harish-Chandra

In mathematics , Borel and Harish-Chandra's theorem is a tenet from the theory of arithmetic groups.

Fundamental area for in (in gray) ${\displaystyle SL(2,\mathbb {Z} )}$ ${\displaystyle SL(2,\mathbb {R} )/SO(2)}$

He says that for a semisimple algebraic group a grid in , it therefore a fundamental region finite volume for the effect of on there. The fundamental domain is compact if every unipotent element in belongs to the radical of . ${\displaystyle G(\mathbb {Z} )}$ ${\displaystyle G(\mathbb {R} )}$ ${\displaystyle G(\mathbb {Z} )}$ ${\displaystyle G(\mathbb {R} )}$ ${\displaystyle G(\mathbb {Z} )}$ ${\displaystyle G(\mathbb {Q} )}$

It follows from the theorem that every arithmetic group is a lattice in the connected component of the one of the surrounding Lie group . In particular, arithmetic groups are finitely generated .

A classic example, known since the 19th century, is with a fundamental domain of finite volume. ${\displaystyle SL(2,\mathbb {Z} )\subset SL(2,\mathbb {R} )}$

literature

• A. Borel, Harish-Chandra. Arithmetic subgroups of algebraic groups. Bull. Amer. Math. Soc. 67 (1961), no.6, 579-583. ( PDF )