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arXiv · 2511.20949

A rigidity theorem for complex Kleinian groups

Richard Canary·Tengren Zhang·Andrew Zimmer
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Abstract

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).

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Richard Canary, Tengren Zhang, Andrew Zimmer. 2025-11-26. A rigidity theorem for complex Kleinian groups. https://arxiv.org/abs/2511.20949

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