In Discrete groups, expanding graphs, and invariant measures (in the notes at the end of Chapter 7), Lubotzky mentions that certain estimates for the number of spanning trees $\kappa(G)$ of a $k$-regular graph $G$ have nontrivial arithmetic consequences such as estimates for the class number of certain interesting function fields, citing an unpublished paper of Sarnak from 1988, Number theoretic graphs, which I can't find anywhere on the internet.

I have heard that the class number for the function field of the modular curve $X(N)$ over $\mathbb{F}_p$ is related to the number of spanning trees in the $(p+1)$-regular supersingular isogeny graph $G(N,p)$, whose vertices are supersingular elliptic curves over $\mathbb{F}_N$ and edges are degree $p$ isogenies. I'm not sure about the details of this, so please correct me if I'm wrong here.

Is this the connection Lubotzky was talking about? If so I must be misunderstanding something because Lubotzky mentions that the combinatorial result which has arithmetic corollaries is the following lower bound due to Alon: $$\beta(k)=k-O\left(k \frac{(\log \log k)^2}{\log k}\right)$$ where $\beta(k)$ is the limit inferior of $\kappa(G)^\frac{1}{|G|}$ over all $k$-regular graphs. On the other hand, I believe that, for supersingular isogeny graphs, as $N$ goes to infinity the girth also goes to infinity (correct me if I'm wrong), and for graphs with girth going to infinity it is known that $\kappa(G)^\frac{1}{|G|}$ approaches the constant $\frac{(k-1)^{k-1}}{(k(k-2))^{\frac{k}{2}-1}}$ (see e.g. Lyons, Russell, Asymptotic enumeration of spanning trees), so Alon's result doesn't tell us anything in this case.

Question 1: Does anyone know where I can find this paper of Sarnak, to help sort this out?

Question 2: Am I right that Alon's estimate doesn't say anything about supersingular isogeny graphs, and if so what is the application of Alon's estimate that Lubotzky had in mind?

In Discrete groups, expanding graphs, and invariant measures (in the notes at the end of Chapter 7), Lubotzky mentions that certain estimates for the number of spanning trees $\kappa(G)$ of a $k$-regular graph $G$ have nontrivial arithmetic consequences such as estimates for the class number of certain interesting function fields, citing an unpublished paper of Sarnak from 1988, Number theoretic graphs, which I can't find anywhere on the internet.

I have heard that the class number for the function field of the modular curve $X(N)$ over $\mathbb{F}_p$ is related to the number of spanning trees in the $(p+1)$-regular supersingular isogeny graph $G(N,p)$, whose vertices are supersingular elliptic curves over $\mathbb{F}_N$ and edges are degree $p$ isogenies. I'm not sure about the details of this, so please correct me if I'm wrong here.

Is this the connection Lubotzky was talking about? If so I must be misunderstanding something because Lubotzky mentions that the combinatorial result which has arithmetic corollaries is the following lower bound due to Alon: $$\beta(k)=k-O\left(k \frac{(\log \log k)^2}{\log k}\right)$$ where $\beta(k)$ is the limit inferior of $\kappa(G)^\frac{1}{|G|}$ over all $k$-regular graphs. On the other hand, I believe that, for supersingular isogeny graphs, as $N$ goes to infinity the girth also goes to infinity (correct me if I'm wrong), and for graphs with girth going to infinity it is known that $\kappa(G)^\frac{1}{|G|}$ approaches the constant $\frac{(k-1)^{k-1}}{(k(k-2))^{\frac{k}{2}-1}}$ (see e.g. Lyons, Russell, Asymptotic enumeration of spanning trees), so Alon's result doesn't tell us anything in this case.

Question 1: Does anyone know where I can find this paper of Sarnak, to help sort this out?

Question 2: Am I right that Alon's estimate doesn't say anything about supersingular isogeny graphs, and if so what is the application of Alon's estimate that Lubotzky had in mind?

Edit: I believe that my claim that the girth of supersingular isogeny graphs go to infinity with $N$ is actually false (I confused with other related graphs). On the other hand, I think it is still true that these graphs converge in the Benjamini-Schramm sense to the $k$-regular tree, so we still have that $\kappa(G)^\frac{1}{|G|}$ approaches $\frac{(k-1)^{k-1}}{(k(k-2))^{\frac{k}{2}-1}}$.