Quasi-arithmeticity of lattices in PO(n,1)

TL;DR

This paper investigates the quasi-arithmetic nature of certain non-arithmetic lattices in PO(n,1), showing that some are quasi-arithmetic while others are not, revealing new distinctions among these lattices.

Contribution

It demonstrates that non-arithmetic lattices from Belolipetsky and Thomson are quasi-arithmetic, contrasting with Gromov and Piatetski-Shapiro lattices, and establishes the existence of non-arithmetic lattices not commensurable with known examples.

Findings

Non-arithmetic lattices from Belolipetsky and Thomson are quasi-arithmetic.

Gromov and Piatetski-Shapiro lattices are not quasi-arithmetic.

Existence of non-arithmetic lattices not commensurable with Gromov--Piatetski-Shapiro lattices.

Abstract

We show that the non-arithmetic lattices in PO(n,1) of Belolipetsky and Thomson (2011), obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and Piatetski-Shapiro are not quasi-arithmetic. A corollary of this is that there are, for all $n\geq 2$, non-arithmetic lattices in PO(n,1) that are not commensurable with the Gromov--Piatetski-Shapiro lattices.