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Maximal representations in lattices of the symplectic group

Jacques Audibert

TL;DR

The paper addresses the existence and abundance of Zariski-dense maximal representations inside lattices of the real symplectic group $\mathrm{Sp}(2n,\mathbb{R})$. It develops a two-step approach: first classify which lattices contain the diagonal image $\phi_n(\mathrm{SL}_2)$ through Galois cohomology and the Brauer group, distinguishing even and odd $n$; then produce Zariski-dense maximal representations by bending maximal diagonal representations along curves using centralizers inside the lattice, combined with Strong Approximation to distinguish mapping class group orbits. The main outcome is that every lattice not widely commensurable with $\mathrm{Sp}(4k+2,\mathbb{Z})$ contains infinitely many $\mathrm{MCG}(S_g)$-orbits of Zariski-dense maximal representations that deform from maximal diagonal ones, and in particular $\mathrm{Sp}(4k,\mathbb{Z})$ contains Zariski-dense surface subgroups. These results advance higher Teichmüller theory and thin subgroup constructions in high-rank Lie groups, linking arithmetic twists, deformation theory, and strong approximation to produce rich families of surface subgroups and maximal representations.

Abstract

We prove that all lattices of Sp(2n,R), except those commensurable with Sp(4k+2,Z) when n=2k+1, contain the image of infinitely many mapping class group orbits of Zariski-dense maximal representation that are continuous deformations of maximal diagonal representations. In particular, we show that Sp(4k,Z) contain Zariski-dense surface subgroups for all k.

Maximal representations in lattices of the symplectic group

TL;DR

The paper addresses the existence and abundance of Zariski-dense maximal representations inside lattices of the real symplectic group . It develops a two-step approach: first classify which lattices contain the diagonal image through Galois cohomology and the Brauer group, distinguishing even and odd ; then produce Zariski-dense maximal representations by bending maximal diagonal representations along curves using centralizers inside the lattice, combined with Strong Approximation to distinguish mapping class group orbits. The main outcome is that every lattice not widely commensurable with contains infinitely many -orbits of Zariski-dense maximal representations that deform from maximal diagonal ones, and in particular contains Zariski-dense surface subgroups. These results advance higher Teichmüller theory and thin subgroup constructions in high-rank Lie groups, linking arithmetic twists, deformation theory, and strong approximation to produce rich families of surface subgroups and maximal representations.

Abstract

We prove that all lattices of Sp(2n,R), except those commensurable with Sp(4k+2,Z) when n=2k+1, contain the image of infinitely many mapping class group orbits of Zariski-dense maximal representation that are continuous deformations of maximal diagonal representations. In particular, we show that Sp(4k,Z) contain Zariski-dense surface subgroups for all k.
Paper Structure (13 sections, 18 theorems, 50 equations)

This paper contains 13 sections, 18 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Quaternion algebras
  4. Galois cohomology sets
  5. Exact sequences
  6. The Brauer group
  7. The diagonal embedding
  8. Compatible cocycles
  9. Forms of $\textrm{Sp}(2n,\mathds{R})$
  10. Deformations of maximal diagonal representations
  11. Centralizer of a diagonal element
  12. Mapping class group orbits
  13. Proof of Theorem \ref{['theoremdiagonal']}

Key Result

Theorem 1.1

For any $k\geq1$, $\textnormal{Sp}(4k,\mathds{Z})$ contains Zariski-dense surface subgroups.

Theorems & Definitions (29)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 2.1.1
  • Proposition 2.2.1: Proposition 1 Chapter III Serre_GaloisCohomology
  • Proposition 2.4.1: Theorem 4.4.7 in Gille_CentralSimpleAlgebrasGaloisCohomology
  • Proposition 2.4.2: Theorem 20 Chapter X Albert_Structureofalgebras
  • Lemma 3.1.1
  • proof
  • Lemma 3.1.2
  • ...and 19 more