On quasi-arithmeticity of hyperbolic gluings

TL;DR

This paper investigates the conditions under which hyperbolic orbifold gluings are quasi-arithmetic, revealing that such gluings impose quasi-arithmeticity on individual building blocks and providing examples that challenge previous assumptions about arithmeticity.

Contribution

It establishes new criteria linking the quasi-arithmeticity of hyperbolic gluings to their components and presents novel examples illustrating these phenomena.

Findings

Quasi-arithmetic hyperbolic gluings require each building block to be quasi-arithmetic.

Existence of arithmetic gluings with incommensurable building blocks despite reflection symmetries.

Construction of nonarithmetic but quasi-arithmetic orbifolds leading to arithmetic orbifolds through specific gluings.

Abstract

We study a more general version of the gluings of hyperbolic orbifolds in the spirit of Gromov and Piatetski-Shapiro, where the gluing pieces, called the building blocks, are no longer assumed to be arithmetic or incommensurable. We prove that if such a general hyperbolic gluing along a common finite-volume totally geodesic hypersurface is quasi-arithmetic (this is a broader notion than that of arithmeticity) then each building block must be quasi-arithmetic as well and, moreover, with the same ambient group and adjoint trace field. We also show that there exist arithmetic gluings whose building blocks are incommensurable even despite the reflection with respect to the lift of the gluing locus commensurates the fundamental group of the gluing. On the other hand, we provide an example of nonarithmetic but quasi-arithmetic orbifolds such that a specific gluing of such an orbifold with itself along the boundary gives rise to an arithmetic hyperbolic orbifold. We illustrate the above results in the setting of reflection groups and hyperbolic Coxeter polyhedra and apply them to rule out the (quasi-)arithmeticity of a family of ideal hyperbolic right-angled $3$-polyhedra, namely, certain ``twisted'' ideal right-angled antiprisms, which play an important role in low-dimensional geometry and topology.