Lemma from Selberg

In mathematics, the Selberg lemma is a fundamental fact about subgroups of the general linear group . (Occasionally the term von Selberg's Lemma is also used for Malcev's Theorem , another fundamental issue about subsets of the general linear group.) It is named after Atle Selberg .

Lemma from Selberg

If is a field of the characteristic , then every finitely generated subgroup of is virtually torsion-free , that is, it contains a torsion-free subgroup of finite index . ${\displaystyle K}$ ${\displaystyle 0}$ ${\displaystyle GL(n,K)}$

Selberg published his proof in 1960; in 1976 JWS Cassels gave a simpler proof, another elementary proof comes from Roger Alperin .

A geometrical interpretation: Every 3-dimensional hyperbolic orbifold is finitely overlaid by a hyperbolic manifold , more generally every locally symmetric orbifold modeled after or is finitely overlaid by a locally symmetric space . ${\displaystyle SL(n,\mathbb {C} )/SU(n)}$ ${\displaystyle SL(n,\mathbb {R} )/SO(n)}$

An example by M. Kapovich shows that the statement of the lemma cannot be generalized to discrete isometric groups of negatively curved manifolds.

example

${\displaystyle SL(2,\mathbb {Z} )}$is a finitely generated subgroup of , for example a generating system is . ${\displaystyle GL(2,\mathbb {R} )}$ ${\displaystyle \left\{A=\left({\begin{array}{cc}0&-1\\1&0\end{array}}\right),B=\left({\begin{array}{cc}1&1\\0&1\end{array}}\right)\right\}}$

${\displaystyle SL(2,\mathbb {Z} )}$is not torsion free, for example is , however are the congruence subsets ${\displaystyle A^{4}=1}$

${\displaystyle \Gamma (N):=\mathrm {ker} (SL(2,\mathbb {Z} )\to SL(2,\mathbb {Z} /N\mathbb {Z} ))}$

Subgroups of finite index and, according to a classical Minkowski theorem (1887), torsion-free for . ${\displaystyle N\geq 3}$

literature

• Ratcliffe, John G .: Foundations of hyperbolic manifolds. Second edition. Graduate Texts in Mathematics, 149. Springer, New York, 2006. ISBN 978-0387-33197-3 ; 0-387-33197-2. (§7.5)

Web links

• Nica: Linear groups - Malcev's theorem and Selberg's lemma

Individual evidence

1. ^ Selberg, Atle: On discontinuous groups in higher-dimensional symmetric spaces. 1960 Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960) pp. 147-164 Tata Institute of Fundamental Research, Bombay
2. ^ Cassels, JWS: An embedding theorem for fields. Bull. Austral. Math. Soc. 14 (1976) no. 2, 193-198.
3. ↑ Alperin, Roger C .: An elementary account of Selberg's lemma. Enseign. Math. (2) 33 (1987) no. 3-4, 269-273.
4. ↑ M. Kapovich: “A note on Selberg's Lemma and negatively curved Hadamard manifolds”, ArXiv