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Klein-Maskit combination theorem for Anosov subgroups: Free products

Subhadip Dey, Michael Kapovich

Abstract

We prove a generalization of the classical Klein-Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if $Γ_A$ and $Γ_B$ are Anosov subgroups of a semisimple Lie group $G$ of noncompact type, then under suitable topological assumptions, the group generated by $Γ_A$ and $Γ_B$ in $G$ is again Anosov, and is naturally isomorphic to the free product $Γ_A*Γ_B$. Such a generalization was conjectured in our previous article with Bernhard Leeb (arXiv:1805.07374).

Klein-Maskit combination theorem for Anosov subgroups: Free products

Abstract

We prove a generalization of the classical Klein-Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if and are Anosov subgroups of a semisimple Lie group of noncompact type, then under suitable topological assumptions, the group generated by and in is again Anosov, and is naturally isomorphic to the free product . Such a generalization was conjectured in our previous article with Bernhard Leeb (arXiv:1805.07374).
Paper Structure (20 sections, 25 theorems, 63 equations, 3 figures)

This paper contains 20 sections, 25 theorems, 63 equations, 3 figures.

Table of Contents

  1. Preliminary notions and results
  2. Reduced forms
  3. Hausdorff distance and convergence
  4. Geometric background
  5. Flag varieties
  6. Contraction and regularity
  7. Pure sequences
  8. A regularity criterion
  9. Regular and Anosov subgroups
  10. Discreteness
  11. Regularity
  12. Boundary embedding
  13. Construction of $\xi$
  14. $\xi$ is antipodal
  15. $\xi$ is an embedding
  16. ...and 5 more sections

Key Result

Theorem A

Suppose that $A, B\subset {\rm Flag}({\tau_{\rm mod}})$ are (disjoint) compact sets with nonempty interiors which are antipodal (see def:antipodal) to each other. Let $\Gamma_A$ and $\Gamma_B$ be ${\tau_{\rm mod}}$-Anosov subgroupsAlthough we do not state original definition of ${\tau_{\rm mod}}$-A Then:

Figures (3)

  • Figure 1:
  • Figure 2: Fellow traveling ray
  • Figure 3:

Theorems & Definitions (52)

  • Theorem A: Combination Theorem
  • Definition 1.1: Antipodality
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof : Proof of \ref{['lem:nontrivial projection']}
  • Corollary 1.4
  • Remark 1.5
  • Proposition 1.6
  • proof
  • ...and 42 more