$\mathbb{F}_2$ is residually finite, but what are the trivially-intersecting subgroups?

Question

A group $G$ is called residually finite if for all $g\in G$ with $g\not=1$ there exists a normal subgroup of finite index, $N_g\lhd_f G$ such that $g\not\in N_g$. Note that $\bigcap\limits_{g\in G} N_g=1$.

It is well-known that $\mathbb{F}_2$ is residually finite. To prove this, simply recall that linear groups are residually finite, and $F_2$ is linear because the matrices, $$\left( \begin{array}{ccc} 1 & 2 \\ 0 & 1 \end{array} \right) \text{ and } \left( \begin{array}{cc} 1 & 0 \\ 2 & 1 \end{array} \right)$$ generate a free group. However, this proof is somewhat unsavoury. I would quite like to know what the subgroups $N_g$ are.

Does there exist a "nice" set of finite-index subgroups of $\mathbb{F}_2$ which intersect trivially?

Nice is, of course, a subjective term. By "nice" I could mean "take the same form". However, I doubt this can happen (if they "take the same form" then presumably some rule dictates this form, and this rule is defined by a word, or a collection of words, and so the intersection of the subgroups is non-trivial). Alternatively, I could mean characteristic, which is nice in a different sense. I suppose if you can given a reason why I might think your set is nice that would be...nice.

Answer 1

Here is one possibility, among many. Fix a prime number $p$. For any group $G$, define $\gamma_{1}^{p}(G) = G$ and, for $n\geq 1$, define $$\gamma_{n+1}^{p}(G) = \left(\gamma_{n}^{p}(G)\right)^{p}[G,\gamma_{n}^{p}(G)],$$ where $[A,B]$ denotes the subgroup generated by commutators of the form $[a,b]$, with $a\in A$ and $b\in B$. If $G$ is finitely generated, then $G/\gamma_{n}^{p}(G)$ is a finite $p$-group, for all $n$. In particular, for the free group $F_{2}$ of rank two, the groups $F_{2}/\gamma_{n}^{p}(F_{2})$ are all finite. Moreover, since free groups are residually $p$-finite, we have $\bigcap_{n\geq 1}\gamma_{n}^{p}(F_{2}) = 1$.

Answer 2

Here is a topological argument:

In Hatcher's book, one may find the following easy lemma (chapter 1.A, exercice 10):

Lemma: Let $X$ be the wedge sum of $n$ circles, with its natural graph structure, and let $\tilde{X} \to X$ be a covering space with $Y \subset \tilde{X}$ a finite connected subgraph. Show there is a finite graph $Z \supset Y$ having the same vertices as $Y$, such that the projection $Y \to X$ extends to a covering space $Z \to X$.

Let $X$ be a bouquet of two circles and $\tilde{X}$ be its universal covering, namely the Cayley graph of $\mathbb{F}_2$. For all $n \geq 1$, let $X_n \to X$ be the covering extending the projection $B(1,n) \to \tilde{X}$ of the ball $B(1,n)$ of radius $n$ and centered at $1$ in the graph $\tilde{X}$; it is easy to see that $X_n$ is just $B(1,n)$ with additional edges between vertices of the sphere $S(1,n)$. Therefore, the non-trivial elements of $\mathbb{F}_2$ in the ball $B(1,n-1)$ are not in $\pi_1(X_n)$, hence $\bigcap\limits_{n \geq 1} \pi_1(X_n) = \{1\}$. Moreover, each $\pi_1(X_n)$ is a finite-index subgroup of $\pi_1(X)= \mathbb{F}_2$ since the graph $X_n$ is finite.

The subgroups $\pi_1(X_n)$ are not described quite explicitely, but it is easy to draw $X_n$ for small $n$, so one may hope to find a generator set for these groups.