Malcev's theorem

As a set of Malcev in is mathematics a fundamental facts about subgroups of the general linear group called.

Malcev's theorem

Every finitely generated subgroup is residual finite , that is, for each there is a homomorphism to a finite group with . (Equivalent: for each there is a subgroup of finite index with .) ${\displaystyle \Gamma \subset GL(n,\mathbb {C} )}$ ${\displaystyle a\in \Gamma \setminus \left\{1\right\}}$ ${\displaystyle \phi \colon \Gamma \rightarrow F}$ ${\displaystyle F}$ ${\displaystyle \phi (a)\not =1}$ ${\displaystyle a\in \Gamma \setminus \left\{1\right\}}$ ${\displaystyle \Gamma ^{\prime }\subset \Gamma }$ ${\displaystyle a\notin \Gamma ^{\prime }}$

This theorem is also called Selberg's Lemma, although it was first proved by Malcev .

A topological interpretation: Let a 3-dimensional hyperbolic manifold be (or more generally a locally symmetrical space modeled after or after ), then for every closed curve there is a finite superposition in which the raised curve is not closed. ${\displaystyle M}$ ${\displaystyle SL(n,\mathbb {C} )/SU(n)}$ ${\displaystyle SL(n,\mathbb {R} )/SO(n)}$ ${\displaystyle \gamma \subset M}$ ${\displaystyle {\widehat {M}}\to M}$ ${\displaystyle {\hat {\gamma }}}$

literature

• A. Malcev: On isomorphic matrix representations of infinite groups. In: Rec. Math. [Mat. Sbornik] NS Volume 8, No. 50, 1940, pp. 405-422. (Russian)

Web links

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