Let $S_{g}$ be the genus $g$ closed orientable surface, and let $\tau \subset S_{g}$ be a connected, generic train track (all switches are trivalent). Let $\mathcal{B}$ denote the number of branches and $\mathcal{S}$ the number of switches.

Label the switches from $1$ to $\mathcal{S}$, and the branches from $1$ to $\mathcal{B}$. We can identify $\mathbb{R}^{\mathcal{B}}$ with the set of real-valued weights on the branch set of $\tau$. Then there is a linear map $L_{\tau}:\mathbb{R}^{\mathcal{B}}\rightarrow \mathbb{R}^{\mathcal{S}}$ associated to $\tau$ which is defined by, given $u\in \mathbb{R}^{\mathcal{B}}$, the $j^{th}$ coordinate of $L_{\tau}(u)$ is the sum of the weights at the two incoming branches of the $j^{th}$ switch, minus the weight at the outgoing branch at this switch.

The map is defined so that the non-negative vectors in the kernel are precisely the transverse measures on $\tau$.

**My question is: What is a lower bound (as a function of $g$) for the smallest, nonzero eigenvalue of $(L_{\tau}L_{\tau}^{T})$?**

Remark (1): Such a bound exists, since there are only finitely many homeomorphism types of train tracks on $S_{g}$, and therefore only finitely many such linear maps.

Remark (2): Technically, I'm only interested in the case when $\tau$ is large, i.e., $S_{g}\setminus \tau$ is simple connected.

Remark (3): Thinking of $\tau$ as a graph, $L_{\tau}$ is the oriented edge-vertex matrix, so what I think I am asking for is a lower bound for the spectral gap of such a graph. However, this graph is not simplicial. It may have loops and double edges so there will be more than one zero eigenvalue, so maybe "spectral gap" is not technically the correct term; it's really the smallest nonzero eigenvalue that I'm interested in. Something like Cheeger's inequality might work, but again I'm not sure how to phrase it in the case of non-simple graphs.

Thank you for reading! Any help would be greatly appreciated!