Residual finite group

Residual finite groups are a term from the mathematical field of group theory . These are (infinite) groups that can be approximated in a certain way by finite groups .

definition

A group is called residual finite if there is a subgroup of finite index for each element other than the neutral element ${\ displaystyle G}$ ${\ displaystyle e_ {G}}$ ${\ displaystyle g \ in G}$

{\ displaystyle H \ subset G}

With

{\ displaystyle g \ not \ in H}

gives. In other words

{\ displaystyle \ bigcap _ {\ left [G \ colon H \ right] <\ infty} H = \ left \ {e_ {G} \ right \}}

,

d. H. the average of all subgroups of finite index consists only of the neutral element .

Is equivalent to the condition that there will be any different from the neutral element member a homomorphism in a finite group with intended to provide. ${\ displaystyle g \ in G}$ ${\ displaystyle \ phi \ colon G \ to K}$ ${\ displaystyle K}$ ${\ displaystyle \ phi (g) \ not = e_ {K}}$

Examples

According to Malcev's theorem , every finitely generated subgroup of the general linear group is residual finite for every commutative ring with one. ${\ displaystyle GL (n, R)}$ ${\ displaystyle R}$

Numerous examples of residual finite groups result from this criterion:

free groups
Area groups
Fundamental groups of compact locally symmetric spaces , especially compact hyperbolic manifolds

Polycyclic and nilpotent groups that are finally generated are residual finite.

Fundamental groups of compact 3-manifolds are residual finite, although in general it is not known whether they are isomorphic to subsets of . ${\ displaystyle GL (n, K)}$

The following also applies:

Subgroups of residual finite groups are again residual finite.
If there is a residual finite subgroup with , then residual is also finite. ${\ displaystyle H \ subset G}$ ${\ displaystyle \ left [G \ colon H \ right] <\ infty}$ ${\ displaystyle G}$

The tree lag solitaire groups are not residual finite.

It is an open question whether there are hyperbolic groups that are not residual finite.

properties

Residual finite groups have an algorithmically solvable word problem .
Residual finite groups have the Hopf property : every epimorphism of the group is an isomorphism .

The following properties of a group are equivalent:

${\ displaystyle G}$ is residual finite.
The canonical mapping into pro-finite completion is injective. ${\ displaystyle {\ hat {G}}}$
The trivial subgroup is separable .
The pro-finite topology is Hausdorffian .

Topological interpretation

The fundamental group of a CW-complex is residual finite if and only if there is a finite cover for every compact subset of the universal cover such that ${\ displaystyle \ pi _ {1} X}$ ${\ displaystyle X}$ ${\ displaystyle K \ subset {\ widetilde {X}}}$ ${\ displaystyle P \ colon {\ widetilde {X}} \ to X}$ ${\ displaystyle p \ colon Y \ to X}$

{\ displaystyle p ^ {- 1} P (K) \ to Y}

is an embedding .

This criterion can be used in different situations to check that immersions can be elevated to embeddings in a finite superposition. It is used, for example, in works on the virtual hook conjecture and in the proof of the Taubes conjecture by Friedl-Vidussi.

Meaning in algebraic geometry

Let it be a scheme of finite type over . Then there is homomorphism ${\ displaystyle X}$ ${\ displaystyle \ mathbb {C}}$

{\ displaystyle \ pi _ {1} ^ {top} (X) \ to \ pi _ {1} ^ {et} (X)}

Injective exactly when residual is finite. ${\ displaystyle \ pi _ {1} ^ {top} (X)}$

literature

W. Magnus: Residually finite groups. Bull. Amer. Math. Soc. 75 (1969) 305-316. on-line

Web links

Residual finiteness
Residual finiteness results
Berstein Seminar: Residual finiteness and Hopfian groups

Individual evidence

^ Hirsch, KA: On infinite soluble groups. IV. J. London Math. Soc. 27, (1952). 81-85.
^ Hempel, John: Residual finiteness for 3-manifolds. Combinatorial group theory and topology (Alta, Utah, 1984), 379-396, Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987.
↑ Malcev, A .: On isomorphic matrix representations of infinite groups. (Russian) Rec. Math. [Mat. Sbornik] NS 8 (50), (1940). 405-422.
^ Scott, Peter: Subgroups of surface groups are almost geometric. J. London Math. Soc. (2) 17 (1978) no. 3, 555-565.
^ Agol, Ian: The virtual hook conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning. Doc. Math. 18 (2013), 1045-1087.
↑ Friedl, Stefan; Vidussi, Stefano: Twisted Alexander polynomials detect fibered 3-manifolds. Ann. of Math. (2) 173 (2011), no. 3, 1587-1643.

[1] Hirsch, KA: On infinite soluble groups. IV. J. London Math. Soc. 27, (1952). 81-85.

[2] Hempel, John: Residual finiteness for 3-manifolds. Combinatorial group theory and topology (Alta, Utah, 1984), 379-396, Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987.

[3] Malcev, A .: On isomorphic matrix representations of infinite groups. (Russian) Rec. Math. [Mat. Sbornik] NS 8 (50), (1940). 405-422.

[4] Scott, Peter: Subgroups of surface groups are almost geometric. J. London Math. Soc. (2) 17 (1978) no. 3, 555-565.

[5] Agol, Ian: The virtual hook conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning. Doc. Math. 18 (2013), 1045-1087.

[6] Friedl, Stefan; Vidussi, Stefano: Twisted Alexander polynomials detect fibered 3-manifolds. Ann. of Math. (2) 173 (2011), no. 3, 1587-1643.