A rigidity theorem for complex Kleinian groups

TL;DR

This paper characterizes when certain complex Kleinian groups are isomorphic to uniform lattices in PSL(2,C) based on their critical exponents, extending understanding of their geometric and algebraic structure.

Contribution

It establishes a precise criterion linking the k-th simple root critical exponent to the isomorphism with uniform lattices in PSL(2,C), and describes the structure of hyperconvex subgroups with critical exponent 2.

Findings

A (d-k)-hyperconvex subgroup is isomorphic to a uniform lattice in PSL(2,C) iff its k-th simple root critical exponent equals 2.

Strongly irreducible hyperconvex subgroups with critical exponent 2 are images of uniform lattices under irreducible representations.

Provides a rigidity criterion connecting critical exponents to the subgroup's algebraic structure.

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).