# Arithmeticity, superrigidity, and totally geodesic submanifolds

Uri Bader, David Fisher, Nicholas Miller and Matthew Stover. Annals of Mathematics 193 (2021), 837–861. DOI 10.4007/annals.2021.193.3.4.

The question is whether infinitely many maximal totally geodesic subspaces of dimension at least two force a real hyperbolic lattice to be arithmetic. Applications concern knot complements, geometric finiteness and Dehn fillings. The superrigidity theorem is an intermediate result with its own hypotheses.

Read the introduction, then Proposition 3.1 (geometric subspaces to homogeneous measures), Proposition 3.4 (lifting measures), Lemma 3.5 (compatibility), the algebraic-representation machinery in Section 4, Theorem 1.6 and the local-place arithmeticity deduction. Read the Dehn-filling argument separately.

[Interactive argument map](argument.roadmap.html) · [arXiv v5 PDF](paper.pdf) · [Published PDF](published-nsf.pdf) · [Expanded native source](manuscript.tex)

The native arXiv v5 source is dated 26 April 2020 and has 23 PDF pages. The journal has 25 article pages, with a separate JSTOR cover in the cached 26-page file. This review freezes the received date, 20 June 2019, as its external-result cutoff. Later correction history and independently proposed new repairs are recorded separately. The arXiv history explicitly records prepublication corrections to Proposition 3.1 and the measure-construction argument; this does not determine which issues remain in the journal.

The map indexes 23 source claim environments and 41 bibliography entries. Fourteen theorem/lemma/proposition units and two prose packets cover the mathematical body. Edges are candidate dependencies, not certificates. Native source numbering and journal pagination need separate checks. External theorem statements are not verified merely by appearing in the bibliography. Mathematical content was not edited: TeX comments were masked with spaces and the packaged bibliography was expanded in place. No manuscript code was executed.