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Quasi-isometric free group representations into SL_3(R)

León Carvajales, Pablo Lessa, Rafael Potrie

Abstract

We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space.

Quasi-isometric free group representations into SL_3(R)

Abstract

We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space.

Paper Structure

This paper contains 25 sections, 23 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main problem
  3. Rank $1$ groups and products
  4. Free groups in $\mathop{\mathrm{\mathsf{SL}}}\nolimits_3(\mathbb{R})$
  5. Non-Anosov Examples
  6. Derived from Barbot representations
  7. Organization of the paper
  8. Acknowledgements
  9. Products of $\mathop{\mathrm{\mathsf{SL}}}\nolimits_2$'s
  10. Robust faithfulness in $\mathop{\mathrm{\mathsf{SL}}}\nolimits_2(\mathbb{R})$
  11. Proof of Theorem \ref{['theoproduct']}
  12. Perturbed not quasi-isometric representations
  13. General properties of quasi-isometric representations
  14. A criterion for non quasi-isometry
  15. Examples of quasi-isometric free groups in $\mathop{\mathrm{\mathsf{SL}}}\nolimits_3(\mathbb{R})$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose that $\Gamma$ is a finitely generated non-abelian free group and $\mathsf{G} = \mathop{\mathrm{\mathsf{SL}}}\nolimits_2(\mathbb{R})^r \times \mathop{\mathrm{\mathsf{SL}}}\nolimits_2(\mathbb{C})^s$ for some non-negative integers $r$ and $s$. Then $\rho:\Gamma \to \mathsf{G}$ is robustly faith

Figures (1)

  • Figure 1: Proof of Lemma \ref{['lem: free generators of subgroups']}. On the left, the case $k=3$. The loop $a$ has two lifts $a_1$ and $a_2$ which are loops, while $b$ lifts to two non-closed paths $b_1$ and $b_2$. The fundamental group $\pi_1(\widetilde{X},\widetilde{x}_0)$ is freely generated by the homotopy classes of the concatenated paths $a_1$, $b_1\star b_2$, and $b_1\star a_2\star \overline{b}_1$ (where $\overline{\beta}$ denotes the path $\beta$ travelled with the opposite orientation). In this case, one may take $p=1$. On the right, the case $k=4$. Each generator in $\{a,b\}$ has two open lifts, and a closed one. A free generator of $\pi_1(\widetilde{X},\widetilde{x}_0)$ is induced by $a_1$, $b_1\star b_2$, $b_1\star a_3\star a_2 \star \overline{b}_1$, and $b_1\star a_3\star b_3 \star \overline{a}_3\star \overline{b}_1$, so $p=2$ in this case.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 34 more