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Greenberg-Shalom's Commensurator Hypothesis and Applications

Nic Brody, David Fisher, Mahan Mj, Wouter van Limbeek

TL;DR

This work investigates the Greenberg-Shalom hypothesis, positing that a discrete Zariski-dense subgroup with almost dense commensurator must be an arithmetic lattice, and systematically derives far-reaching consequences across products of Lie groups, $p$-adic groups, and trees. By formalizing irreducibility and connecting it to commensurators, the authors show that under the hypothesis many irreducible subgroups are forced to be lattices, yielding corollaries about coherence, Margulis-Zimmer-type classifications, and arithmeticity in hyperbolic manifolds. The paper extends the scope to automorphism groups of trees, improving FLSS-type results and ruling out certain irreducible subgroups in product settings, while also addressing rank phenomena by proving that irreducible lattices in multi-factor products have arbitrarily small generating sets, with implications for rank gradients and cost. Overall, the results provide a cohesive framework linking rigidity, arithmeticity, and geometric group theory through the Greenberg-Shalom lens, with conditional statements awaiting resolution in full generality.

Abstract

We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the $`70$s and more generally by Shalom in the early $2000$s. We refer to this positive answer as the Greenberg-Shalom hypothesis. This hypothesis then says that any infinite discrete subgroup of a semisimple Lie group with dense commensurator is a lattice in a product of some factors. For some applications it is natural to extend the hypothesis to cover semisimple algebraic groups over other fields as well.

Greenberg-Shalom's Commensurator Hypothesis and Applications

TL;DR

This work investigates the Greenberg-Shalom hypothesis, positing that a discrete Zariski-dense subgroup with almost dense commensurator must be an arithmetic lattice, and systematically derives far-reaching consequences across products of Lie groups, -adic groups, and trees. By formalizing irreducibility and connecting it to commensurators, the authors show that under the hypothesis many irreducible subgroups are forced to be lattices, yielding corollaries about coherence, Margulis-Zimmer-type classifications, and arithmeticity in hyperbolic manifolds. The paper extends the scope to automorphism groups of trees, improving FLSS-type results and ruling out certain irreducible subgroups in product settings, while also addressing rank phenomena by proving that irreducible lattices in multi-factor products have arbitrarily small generating sets, with implications for rank gradients and cost. Overall, the results provide a cohesive framework linking rigidity, arithmeticity, and geometric group theory through the Greenberg-Shalom lens, with conditional statements awaiting resolution in full generality.

Abstract

We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the s and more generally by Shalom in the early s. We refer to this positive answer as the Greenberg-Shalom hypothesis. This hypothesis then says that any infinite discrete subgroup of a semisimple Lie group with dense commensurator is a lattice in a product of some factors. For some applications it is natural to extend the hypothesis to cover semisimple algebraic groups over other fields as well.
Paper Structure (21 sections, 32 theorems, 23 equations)

This paper contains 21 sections, 32 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Notation, standing assumptions, and definitions of irreducibility
  3. Irreducible groups are lattices
  4. Irreducible groups in products of semisimple Lie groups
  5. Groups generated by parabolic elements
  6. Arithmetic of hyperbolic 3-manifolds
  7. Coherence
  8. Margulis-Zimmer conjecture
  9. Automorphism groups of trees
  10. On a theorem from FLSS
  11. Proof of Corollaries \ref{['noirreduciblesurfacesinpadics']} and \ref{['cor:algebraicsurfacegroups']}
  12. Rank gradient and small generating sets
  13. Historical remarks and motivation
  14. The Greenberg-Shalom hypothesis and rank
  15. Proofs of Lemma \ref{['lemma:Z-generation']} and Lemma \ref{['lemma:independence']}
  16. ...and 6 more sections

Key Result

Lemma 2.9

Any discrete irreducible subgroup $\Theta\subseteq G$ has unbounded projections to every simple factor $G_i$. Moreover the set of elements projected to any bounded open set in $G/G_i$ is already unbounded in $G_i$.

Theorems & Definitions (81)

  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9
  • proof
  • Proposition 2.10
  • Lemma 2.11
  • ...and 71 more