Lower bound on growth for intersection of two subgroups of free group Let $\Gamma_1$ and $\Gamma_2$ be two subgroups of the rank-$2$ free group $F_2$. Can then one find a nontrivial lower bound on the growth exponent of their intersection $\Gamma_1 \cap \Gamma_2$, in terms of the growth exponents of the two subgroups?
Here by the growth exponent of a subgroup $\Gamma \subset F_2$ I mean the real number
$$\lim_{n \to \infty} \frac{1}{n} \log \# \{w \in \Gamma \;\mid\; \ell(w) \leq n\},$$
where $\ell$ denotes the word length inside $F_2$; so that this exponent lies between $0$ and $\log 3$.
For semigroups, clearly there is no hope for such a lower bound. Indeed, in the rank-$2$ free monoid $A^*$ (on the alphabet $A = \{a, b\}$), the two subsemigroups $S_1 := (aA^*)^+$ and $S_2 := (bA^*)^+$ (generated by all the words starting with $a$, resp. with $b$) are disjoint, but have maximal growth exponent (namely $\log 2$).
However this construction does not work at all for groups. I have tried to find pairs of subgroups in $F_2$ that have large growth exponents but trivial (or at least small) intersection, but I am not even sure where to start. Any help would be appreciated!
 A: OK, in fact I figured this out as soon as I properly understood how Stallings graphs work. Here is the analogous construction for groups.
Let $\Gamma_0$ be the subgroup of the free group $F_2 = \langle a, b \rangle$ generated by the set $\{a^{k+1} b^{-1} a^{-k} \;\mid\; k \geq 0\}$. I claim that elements of $\Gamma_0$ can be characterized as follows: a reduced word $w(a,b,a^{-1},b^{-1})$ lies in $\Gamma_0$ if and only if the word obtained from $w$ by substituting an opening parenthesis to each $a$ and $b$, and a closing parenthesis to each $a^{-1}$ and $b^{-1}$, is well-parenthesized.
This is probably easy to show by hand, but the right way to see things is probably to use Stallings graphs (see e.g. these slides for a very quick visual introduction: http://www.lix.polytechnique.fr/combi/archivesSeminaire/transparents/150513_Frederique_Bassino.pdf, or this paper for a more detailed one: https://arxiv.org/abs/math/0202285 ). Their key property is that a word $w \in F_2$ belongs to some subgroup $\Gamma$ if and only if there exists a loop in the Stallings graph of $\Gamma$, starting and ending at its basepoint, and labeled by $w$. Now the Stallings graph of $\Gamma_0$ has vertices indexed by $\mathbb{Z}_{\geq 0}$ (with basepoint $0$) and, for each vertex $n$, an edge labeled $a$ and an edge labeled $b$ both going from $n$ to $n+1$; and the characterization follows immediately.
It is then not very hard to see that $\Gamma_0$ has maximal possible growth exponent, namely $\log 3$. In fact, the number of words of length $2n$ in $\Gamma$ is twice the sequence https://oeis.org/A059231 , which grows like a polynomial times $3^{2n}$ (OEIS gives the asymptotic estimate "a(n) ~ sqrt(2)3^(2n+1)/(8*sqrt(Pi)*n^(3/2))").
Now we set $\Gamma_1 := a \Gamma_0 a^{-1}$ and $\Gamma_2 := b \Gamma_0 b^{-1}$. Of course these two groups still have growth exponent $\log 3$. However, their intersection is trivial: indeed, all nontrivial elements of $\Gamma_1$ (resp. of $\Gamma_2$) actually start with $a$ (resp. $b$) and end with $a^{-1}$ (resp. $b^{-1}$), even when written in reduced form.
This last statement is obvious from the characterization of $\Gamma_0$, but once again it is helpful to think in terms of Stallings graphs, in particular so as to bring into light the parallel with the semigroup construction. The Stallings graph of $\Gamma_1$ (resp. $\Gamma_2$) is obtained from the Stallings graph of $\Gamma_0$ by adding a new basepoint, with a single edge from that vertex to the old basepoint labeled by $a$ (resp. by $b$).
