Vign\'eras orbifolds: isospectrality, regulators, and torsion homology

TL;DR

This paper introduces new criteria for isospectrality of Vignéras orbifolds, enabling the construction of small exotic examples and exploring regulator quotients and torsion homology relations.

Contribution

It develops a novel approach to analyze isospectrality, representation equivalence, and regulator quotients in Vignéras orbifolds, producing explicit examples and linking torsion homology with Galois representations.

Findings

Constructed small exotic hyperbolic 3-orbifolds with specific isospectral properties.

Established criteria for regulator quotient rationality even when orbifolds are not isospectral.

Linked primes in regulator quotients to torsion homology differences and Galois representations.

Abstract

We develop a new approach to the isospectrality of the orbifolds constructed by Vign\'eras. We give fine sufficient criteria for i-isospectrality in given degree i and for representation equivalence. These allow us to produce very small exotic examples of isospectral orbifolds: hyperbolic 3-orbifolds that are i-isospectral for all i but not representation equivalent, hyperbolic 3-orbifolds that are 0-isospectral but not 1-isospectral, and others. Using the same method, we also give sufficient criteria for rationality of regulator quotients Reg_i(Y_1)^2/Reg_i(Y_2)^2 for Vign\'eras orbifolds Y_1, Y_2, sometimes even when they are not isospectral. Moreover, we establish a link between the primes that enter in these regulator quotients and at which torsion homology of Y_1 and Y_2 can differ, and Galois representations.