Profinite recognition of uniformity

Holger Kammeyer

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arXiv · 2610.17041

Profinite recognition of uniformity

Holger Kammeyer

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Abstract

Let $G$ be a semisimple Lie group that is not isogenous to the isometry group of real or complex hyperbolic $n$-space. We show that the rational rank is a profinite invariant for irreducible lattices in $G$, unless $G$ has type $D_{2n}$ which forms a genuine exception. In particular, profinitely isomorphic lattices in $G$ of type $D_{2n}$ can have differing virtual cohomological dimension. As another corollary, uniformity is a profinite property for irreducible lattices in $G$ outside type $D_4$. Assuming CSP, the existence of triality symmetries gives uniform and non-uniform profinitely isomorphic lattices in type $D_4$.

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BibTeX RIS

Holger Kammeyer. 2026-10-12. Profinite recognition of uniformity. https://arxiv.org/abs/2610.17041

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