The equality conjecture for the volume of the even-dimensional Eisenstein orbifold

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Let n∈2Z≥1n \in 2\mathbb{Z}_{\geq 1}. Write Γeisn∖RHn\Gamma^n_{\mathrm{eis}} \setminus \mathbb{R} H^n for the non-arithmetic hyperbolic orbifold M(Leisn,α0)M({\mathscr{L}}_{\mathrm{eis}}^n,\alpha_0), and let PO⁡(Ψin,Z)∖RHn\operatorname{PO}(\Psi_i^n,\mathbb{Z}) \setminus \mathbb{R} H^n be the associated arithmetic hyperbolic orbifolds. The equality conjecture. The bound in equation is an equality; equivalently,

Vol⁡(Γeisn∖RHn)=∑i=0nVol⁡(PO⁡(Ψin,Z)∖RHn).\operatorname{Vol}\left(\Gamma^n_{\mathrm{eis}} \setminus \mathbb{R} H^n\right)=\sum_{i=0}^n\operatorname{Vol}\left(\operatorname{PO}(\Psi_i^n,\mathbb{Z}) \setminus \mathbb{R} H^n\right).

This predicts that, in every even dimension, the volume of the non-arithmetic hyperbolic orbifold is exactly the sum of the volumes of the associated arithmetic orbifolds. The source gives no resolution of the expected equality.

References

Primary source

Olivier de Gaay Fortman, “Non-arithmetic hyperbolic orbifolds attached to unitary Shimura varieties”, arXiv:2301.01598 (2024).

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