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An Exploration of Compatibility
Leiner, Leah
Leiner, Leah
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Thesis/Dissertation
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2025-08
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Mathematics
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https://doi.org/10.34944/j6q1-2986
Abstract
Let $G=\SO^\circ(3,1)$, $H=\SO^\circ(4,1)$, $\Gamma<G$ be a lattice, and $\rho:\Gamma\to H$ be a Zariski-dense representation. This thesis considers conditions required for $\rho$ to extend to a continuous homomorphism $\widetilde{\rho}:G\to H$, formulated as a classic superrigidity theorem. These conditions require $\Gamma$ to contain an infinite family $\{\Gamma_i\}$ of nonconjugate maximal Fuchsian subgroups associated with closed $\SO^\circ(2,1)$-orbits on $\Gamma\backslash G$ such that $\rho(\Gamma_i)$ fails to be Zariski-dense in $H$. This extends work of Bader--Fisher--Miller--Stover, who used similar techniques find a geometric characterization of arithmeticity. Their work relies on $G$ and $H$ satisfying a technical compatibility condition that does not hold for the pair $G, H$ considered here.
In particular, the situation considered here falls outside the scope covered by their work. This is a interesting case, because $G$ and $H$ are the isometry groups of hyperbolic 3- and 4-space respectively.
This thesis applies a construction of Bader--Furman to the map that $\rho$ will extend to, called $\phi$, and upgrades $\phi$ from a map to a homomorphism, while utilizing a subgroup $J<H$, related to the failure of compatibility. The first goal of this thesis is to understand the failure of compatibility in terms of the triples $(G,H,J)$ with $J<H$ a real algebraic subgroup. A triple is \textit{compatible} or \textit{incompatible} depending on whether the methods of Bader--Fisher--Miller--Stover apply to prove superrigidity via the techniques of Bader--Furman.
We then split into two cases: when $(G,H,J)$ is a compatibility triple and when $(G,H,J)$ is an incompatibility triple. In the former case, we use the methods of Bader--Fisher--Miller--Stover mentioned above to prove superrigidity. In the latter case, we prove there is only one such $J$ that makes an incompatibility triple. In the incompatible case, we use works of Kim--Oh to understand the boundary map $\partial\hh^3\to\partial\hh^4$ classically associated with the representation $\rho$, and this ultimately allows us to conclude that no such $\rho$ could ever exist.
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