LERF

In the mathematical field of group theory , a group is referred to as LERF ( locally extended residually finite , also: subgroup separable ) if there is a subgroup of finite index for every finitely generated subgroup and every element not lying in , which , but does not contain. ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle g}$ ${\displaystyle H}$ ${\displaystyle g}$

The term is particularly important in the low-dimensional topology. There the LERF property of fundamental groups is typically exploited in order to raise the image of an immersion to an embedding in a suitable finite consideration. In connection with the virtual hook conjecture proved by Ian Agol in 2012 , fundamental groups with this property are useful, because then each subgroup that is isomorphic to a surface group corresponds to a surface embedded in a finite consideration.

definition

A group is LERF if there is a homomorphism for each finitely generated subgroup and each ${\displaystyle H\subset G}$ ${\displaystyle g\in G}$

${\displaystyle \phi \colon G\to F}$

from to a finite group there such that and . ${\displaystyle G}$ ${\displaystyle F}$ ${\displaystyle \phi (h)=e\ \forall h\in H}$ ${\displaystyle \phi (g)\not =e}$

An equivalent formulation is that for every finitely generated subgroup the equation ${\displaystyle H}$

${\displaystyle H=\bigcap \left\{G^{\prime }\subset G\colon \left[G\colon G^{\prime }\right]<\infty ,H\subset G^{\prime }\right\}}$

holds that for every element there is a , but not , containing subgroup of finite index . ${\displaystyle g\not \in H}$ ${\displaystyle H}$ ${\displaystyle g}$ ${\displaystyle G^{\prime }\subset G}$

Another equivalent formulation is that every finitely generated subgroup is closed with respect to the pro-finite topology .

Topological interpretation

The fundamental group of a CW complex is LERF if and only if: ${\displaystyle X}$

For each overlay with finitely generated fundamental group and every finite subcomplex there is of layered finite superposition , so the figure a embedding is. ${\displaystyle X^{\prime }\to X}$ ${\displaystyle \Delta \subset X^{\prime }}$ ${\displaystyle X^{\prime }}$ ${\displaystyle {\hat {X}}\to X}$ ${\displaystyle \Delta \to {\hat {X}}}$

Examples

• Abelian groups are LERF.
• Free groups are LERF.
• Surface groups and fundamental groups of Seifert fibers are LERF.
• Fundamental groups of closed hyperbolic 3-manifolds are LERF.
• Free products and more generally products amalgamated along a cyclic group from LERF groups are LERF again.
• Limes groups are LERF.

properties

• Groups that are LERF are also residual finite .
• In groups that are LERF, the word problem is solvable.

literature

• Peter Scott: Subgroups of surface groups are almost geometric . J. London Math. Soc. 17 (1978)
• Rita Gitik: Doubles of groups and hyperbolic LERF 3-manifolds . Ann. of Math. (2) 150 (1999) no. 3, 775-806.

Individual evidence

1. ↑ Lemma 1.4 in: Scott, op. Cit.
2. ^ Marshall Hall : A topology for free groups and related groups. Ann. of Math. (2) 52, (1950). 127-139.
3. ^ G. Peter Scott : Subgroups of surface groups are almost geometric , J. London Math. Soc. 1978, 17: 555-565; Correction: ibid. 32: 217-220 (1985)
4. ^ Ian Agol : The virtual hook conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning. Doc. Math. 18 (2013), 1045-1087.
5. ^ Brunner, Burns, Solitar: The subgroup separability of free products of two free groups with cyclic amalgamation. Contributions to group theory, 90-115, Contemp. Math., 33, Amer. Math. Soc., Providence, RI, 1984.
6. ^ Henry Wilton : Hall's theorem for limit groups. Geom. Funct. Anal. 18 (2008), no. 1, 271-303.