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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.07240 |
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| _version_ | 1866929415783448576 |
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| author | Bartel, Alex Page, Aurel |
| author_facet | Bartel, Alex Page, Aurel |
| contents | We develop a new approach to the isospectrality of the orbifolds constructed by Vignéras. 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 Vignéras 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07240 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vignéras orbifolds: isospectrality, regulators, and torsion homology Bartel, Alex Page, Aurel Number Theory Geometric Topology Representation Theory We develop a new approach to the isospectrality of the orbifolds constructed by Vignéras. 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 Vignéras 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. |
| title | Vignéras orbifolds: isospectrality, regulators, and torsion homology |
| topic | Number Theory Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/2407.07240 |