Non-arithmetic hyperbolic orbifolds attached to unitary Shimura varieties

TL;DR

This paper introduces a novel method to construct non-arithmetic hyperbolic orbifolds in higher dimensions by using anti-holomorphic involutions on complex arithmetic ball quotients, enabling explicit volume calculations.

Contribution

It presents a new construction technique for non-arithmetic lattices in $ ext{PO}(n,1)$ for all $n > 1$, expanding the known examples of such orbifolds.

Findings

Constructed non-arithmetic hyperbolic orbifolds in all dimensions greater than one.

Developed a method to explicitly compute their volumes.

Established a link between anti-holomorphic involutions and hyperbolic orbifold structures.

Abstract

We develop a new method of constructing non-arithmetic lattices in the projective orthogonal group $\text{PO}(n,1)$ for every integer $n$ larger than one. The technique is to consider anti-holomorphic involutions on a complex arithmetic ball quotient, glue their fixed loci along geodesic subspaces, and show that the resulting metric space carries canonically the structure of a complete real hyperbolic orbifold. The volume of various of these non-arithmetic orbifolds can be explicitly calculated.