$S$-arithmetic groups acting simply transitively on products of Bruhat-Tits trees

Abstract

Lubotzky asked when a transitive $\mathfrak{p}$-arithmetic action on a Bruhat--Tits tree arising from a totally definite quaternion algebra can be refined to a simply transitive action. We answer this question affirmatively for class number one maximal orders $\mathcal{O}$ in totally definite quaternion algebras $B$ over number fields $K$, using the classification of Kirschmer and Voight. For all but finitely many prime ideals $\mathfrak{p}$, we construct a congruence arithmetic lattice, commensurable with $\mathcal{O}[1/\mathfrak{p}]^\times/\mathcal{O}_K[1/\mathfrak{p}]^\times$, acting simply transitively on the vertices of the associated Bruhat--Tits tree. We further prove an $S$-arithmetic analogue for products of Bruhat--Tits trees, building on the work of Rungtanapirom, Stix, and Vdovina. These simply transitive actions produce many new examples of Ramanujan Cayley graphs and Ramanujan Cayley regular cubical complexes arising from totally definite quaternion algebras.