A rigidity theorem for complex Kleinian groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914218279698432 |
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| author | Canary, Richard Zhang, Tengren Zimmer, Andrew |
| author_facet | Canary, Richard Zhang, Tengren Zimmer, Andrew |
| contents | Farre, Pozzetti and Viaggi proved that any (d-k)-hyperconvex subgroup of PSL(d,C) is virtually isomorphic to a convex cocompact Kleinian group and that its k-th simple root critical exponent is at most 2. We show that a (d-k)-hyperconvex subgroup is isomorphic to a uniform lattice in PSL(2,C) if and only if its k-th simple root critical exponent is exactly 2. Furthermore, we show that if a strongly irreducible (d-k)-hyperconvex subgroup has k-th simple root critical exponent 2, then it is the image of a uniform lattice in PSL(2, C) by an irreducible representation of PSL(2, C) into PSL(d, C). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_20949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A rigidity theorem for complex Kleinian groups Canary, Richard Zhang, Tengren Zimmer, Andrew Differential Geometry Group Theory 37A17, 22F10, 22E40, 53C30 Farre, Pozzetti and Viaggi proved that any (d-k)-hyperconvex subgroup of PSL(d,C) is virtually isomorphic to a convex cocompact Kleinian group and that its k-th simple root critical exponent is at most 2. We show that a (d-k)-hyperconvex subgroup is isomorphic to a uniform lattice in PSL(2,C) if and only if its k-th simple root critical exponent is exactly 2. Furthermore, we show that if a strongly irreducible (d-k)-hyperconvex subgroup has k-th simple root critical exponent 2, then it is the image of a uniform lattice in PSL(2, C) by an irreducible representation of PSL(2, C) into PSL(d, C). |
| title | A rigidity theorem for complex Kleinian groups |
| topic | Differential Geometry Group Theory 37A17, 22F10, 22E40, 53C30 |
| url | https://arxiv.org/abs/2511.20949 |