Recently, I'm trying to learn geodesic currents on free groups through a paper by Kapovich and Lustig. Let $\langle, \rangle: \overline{CV}(F_N)\times Curr(F_N)\to\mathbb{R}$ be the intersection form and $I$ be a bipartite graph where vertices are $\overline{CV}(F_N)\cup\mathbb{P}Curr(F_N)$ and $[T]\in\overline{CV}(F_N)$ and $[\mu]\in\mathbb{P}Curr(F_N)$ are connected if $\langle [T],[\mu]\rangle=0$. We call $I$ the intersection graph. It is said that many pairs $([T],[\mu])$ form a single edge connected component and I had trouble constructing them ([T] is point in the closure of Outer space i.e. a projective class of $\mathbb{R}$-tree equipped with very small action from $F_N$).
My question hence is:
- Find pairs $([T],[\mu])$ such that they form a single edge connected component.
To answer question 1, here is something random I want to try: consider a hyperbolic surface $S$ with non-empty boundary and with negative Euler characteristic where $\pi_1(S)=F_N$. Let $(\mathfrak{L},\mu)$ be a measured lamination on $S$ and $\widetilde{\mathfrak{L}}$ its lift in $\widetilde{S}$. Then it induced an algebraic lamination $L\subset\partial^2 F_N$ with a current $\mu'$. Let $T_{\mathfrak{L}}$ be the canonical dual tree by associating every non-boundary leaf of $\widetilde{\mathfrak{L}}$ a point of $T_{\mathfrak{L}}$ which is not a branch point, and to the closure of any complementary component of $\widetilde{\mathfrak{L}}$ in $\widetilde{S}$ a branch point. The metric on $T_{\mathfrak{L}}$ is defined as $d(x,y)=\mu(\alpha)$ where $\alpha$ is an arc in $\tilde{S}$ with endpoints on leaves $\ell_x,\ell_y\in\widetilde{\mathfrak{L}}$ respectively. I want to verify if $T_{\mathfrak{L}}, \mu'$ is the pair I'm looking for in question (1). Here comes with my second question:
- How to compute $\langle T_{\mathfrak{L}}, \mu'\rangle$? Is there any geometric meaning of this specific intersection form $\langle T_{\mathfrak{L}}, \mu'\rangle$? Would that be helpful to construct single edge connected component in $I$?
Here is where I'm stuck on question (2): Let $\lambda_i\eta_{g_i}$ be the rational currents converging to $\mu'$. Then $\langle T_{\mathfrak{L}}, \mu'\rangle=\lim_{n\to\infty}\lambda_i\mu(\alpha_i)$ where $\alpha_i$ is the arc realizing translation length of $g_i$ on $T_{\mathfrak{L}}$. I have no other ideas proceeding from here.
I apologized if the question is nonsense and any thoughts would be appreciated!