Residual Finiteness

Residually finite: For any nontrivial element $g\in G$, there is a subgroup $G_1$ of finite index in $G$ which does not contain $g$.

Locally extended residually finite (LERF): If for each finitely generated subgroup $H$ of $G$, for any element $g\in G - H$, there is a subgroup $G_1$ of finite index in $G$ which contains $H$ but not $g$.

Theorem A. Let $X$ be a manifold possibly with boundary with a regular covering $\tilde{X}$ and covering group $G$. Then TFAE:

(1) $G$ is residually finite.

(2) If $C\subset\tilde{X}$ is a compact subset, then the projection map $\tilde{X}\to X$ factors through a finite covering $X_1$ of $X$ such that $C$ projects by a homeomorphism into $X_1$.

Theorem B. Let $X$ be a manifold possibly with boundary with a regular covering $\tilde{X}$ and a covering group $G$. Then TFAE:

(1) $G$ is LERF.

(2) Given a finitely generated subgroup $H$ of $G$ and a compact subset $C$ of $\tilde{X}/H$, there is a finite covering $X_1$ of $X$ such that the projection $\tilde{X}/H\to X$ factors through $X_1$ and $C$ projects homeomorphically into $X_1$.

위의 theorem B는 특히 중요한데, 만약 $\pi_1(M)$이 surface group $H$를 포함하고 있고, LERF라면, $M$이 virtually Haken임을 내포한다. 다시 말해서, surface group을 representing하는 immersed surface in $M$이 적절한 finite covering을 취하면, embedding으로 lift가 된다는 것.

자명하게 LERF는 RF보다 강한 조건이다. Theorem A,B는 LERF와 RF의 기하학적인 의미를 담고 있다. 보통 해석할 때, $\tilde{X}$는 universal cover를 염두해둔다. 이 경우, Residual finiteness는 다음과 같이 해석된다:

$\pi_1(X)$ is residually finite if and only if for every compact subset $C$ of $\tilde{X}$, there is some finite cover $X'\to X$ with $C$ projects homeomorphically.

만약 $X$에 어떤 geometric structure가 있다고 한다면, $X$의 sequence of finite covering $\tilde{X}_i$가 있어서, 점점더 그것의 universal cover $\tilde{X}$에 가까워진다, 수학적으로는 Gromov-Hausdorff converge한다고 볼 수 있다. Hyperbolic 3-manifold에서는 이것을 geometric convergence라고 부른다.

Examples

1. $M$: a Seifert fibered 3-manifold then $\pi_1(M)$ is LERF.

2. $M$: a hyperbolic 3-manifold then $\pi_1(M)$ is LERF. (Virtual Haken/Fibered Conjecture)