LPS-Type Ramanujan Graphs from Definite Quaternion Algebras over $\mathbb Q$ of Class Number One

Abstract: In this paper we construct explicit LPS-type Ramanujan graphs from each definite quaternion algebra over $\mathbb Q$ of class number 1, extending the constructions of Lubotzky, Phillips, Sarnak, and later Chiu, and answering in the affirmative a question raised by Jo and Yamasaki. We do this by showing that for each definite quaternion algebra $\mathcal H$ over $\mathbb Q$ of class number 1 with maximal order $\mathcal O$, if $G = \mathcal H^\times/Z(\mathcal H^\times)$ and $p$ is prime such that $G(\mathbb Q_p) \cong PGL_2(\mathbb Q_p)$, then there exists a congruence $p$-arithmetic subgroup of $G$ which acts simply transitively on the Bruhat-Tits tree of $G(\mathbb Q_p)$.