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Discrete subgroups of semisimple Lie groups, beyond lattices

Fanny Kassel

Abstract

Discrete subgroups of SL(2,R) are well understood, and classified by the geometry of the corresponding hyperbolic surfaces. Discrete subgroups of higher-rank semisimple Lie groups, such as SL(n,R) for n>2, remain more mysterious. While lattices in this setting are rigid, there also exist more flexible, "thinner" discrete subgroups, which may have large and interesting deformation spaces, giving rise in particular to so-called higher Teichmüller theory. We survey recent progress in constructing and understanding such discrete subgroups from a geometric and dynamical viewpoint.

Discrete subgroups of semisimple Lie groups, beyond lattices

Abstract

Discrete subgroups of SL(2,R) are well understood, and classified by the geometry of the corresponding hyperbolic surfaces. Discrete subgroups of higher-rank semisimple Lie groups, such as SL(n,R) for n>2, remain more mysterious. While lattices in this setting are rigid, there also exist more flexible, "thinner" discrete subgroups, which may have large and interesting deformation spaces, giving rise in particular to so-called higher Teichmüller theory. We survey recent progress in constructing and understanding such discrete subgroups from a geometric and dynamical viewpoint.
Paper Structure (55 sections, 9 theorems, 23 equations, 9 figures, 1 table)

This paper contains 55 sections, 9 theorems, 23 equations, 9 figures, 1 table.

Table of Contents

  1. Discrete subgroups of semisimple Lie groups, beyond lattices
  2. Introduction
  3. Lattices
  4. Geometric interpretation
  5. Examples
  6. Rank one versus higher rank
  7. Deformations and rigidity
  8. A change of paradigm
  9. Examples in real rank one
  10. Schottky groups
  11. Quasi-Fuchsian groups
  12. Deformations of Fuchsian representations for higher-dimensional groups
  13. Ping pong in higher real rank
  14. Ping pong in projective space
  15. Schottky groups with disjoint ping pong domains
  16. ...and 40 more sections

Key Result

Lemma 4.3.6

For any $m\geq 2$, there exist $2m$ symplectic simplices $B_1^{\pm}$, $\dots, B_m^{\pm}$ in $\mathbb{P}(\mathbb{R}^{2n})$ such that $B\subset {B'}^*$ for all $B\neq B'$ in $\{B_1^{\pm}, \dots, B_m^{\pm}\}$.

Figures (9)

  • Figure 1: Fundamental domains for the action of $\mathrm{SL}(2,\mathbb{Z})$ on the upper half plane model of $\mathbb{H}^2$
  • Figure 2: A $\delta$-thin triangle in a geodesic metric space. The side $[a,b]$ is contained in the union of the uniform $\delta$-neighbourhoods (indicated by dashes) of the sides $[b,c]$ and $[c,a]$.
  • Figure 3: The limit set (an invariant topological circle) of a quasi-Fuchsian group in $\partial_{\infty}\mathbb{H}^3 \simeq \mathbb{C}\cup\nolinebreak\{\infty\}$
  • Figure 4: Left (resp. right) panel: the dynamics of a large power of $g$ (resp. $g^{-1}$) on $\mathbb{P}(\mathbb{R}^d)$ for a biproximal element $g\in\mathrm{SL}(d,\mathbb{R})$
  • Figure 5: A ping pong configuration as in Claim \ref{['claim:ping-pong-proj']}
  • ...and 4 more figures

Theorems & Definitions (56)

  • Definition 4.2.1
  • Example 4.2.2
  • Example 4.2.3
  • Definition 4.2.4
  • Remark 4.3.1
  • Claim 4.3.2
  • proof
  • Remark 4.3.3
  • Claim 4.3.5
  • proof
  • ...and 46 more