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Anosov representations of amalgams

Subhadip Dey, Konstantinos Tsouvalas

TL;DR

The paper develops a comprehensive framework for constructing Anosov representations of amalgams and HNN extensions of rank-one lattices into higher-rank Lie groups. Central to the approach is the Virtual Amalgam Theorem, which uses antipodal limit sets and interactive pairs/triples in flag varieties to realize amalgams as $oldsymbol{ extTheta}$-Anosov subgroups after passing to suitable finite-index subgroups. It then applies faithful linear embeddings and exterior powers to promote $oldsymbol{1}$-Anosov data to $oldsymbol{2}$-Anosov representations, yielding new examples of one-ended hyperbolic groups that are linear and admit Anosov embeddings in higher rank but do not admit discrete faithful representations into any rank-one group. The paper also extends these methods to rank-one compatible pairs, divides subgroups, and automorphism fixed-point subgroups, producing a broad array of $oldsymbol{2}$-Anosov doubles and HNN extensions, including new instances among $oldsymbol{O}(n,1)$, $oldsymbol{U}(n,1)$, $oldsymbol{Sp}(n,1)$, and $oldsymbol{F}_4^{-20}$. Overall, the work establishes a robust mechanism to generate higher-rank Anosov subgroups from rank-one lattices, enriching higher Teichmüller theory with numerous explicit constructions and applications.

Abstract

For uniform lattices $Γ$ in rank 1 Lie groups, we construct Anosov representations of virtual doubles of $Γ$ along certain quasiconvex subgroups. We also show that virtual HNN extensions of these lattices over some cyclic subgroups admit Anosov embeddings. In addition, we prove that for any Anosov subgroup $Γ$ of a real semisimple linear Lie group $\mathsf{G}$ and any infinite abelian subgroup $\mathrm{H} $ of $ Γ$, there exists a finite-index subgroup $Γ' $ of $ Γ$ containing $\mathrm{H}$ such that the double $Γ' *_{\mathrm{H}} Γ'$ admits an Anosov representation, thereby confirming a conjecture of [arXiv:2112.05574]. These results yield numerous examples of one-ended hyperbolic groups that do not admit discrete and faithful representations into rank 1 Lie groups but do admit Anosov embeddings into higher-rank Lie groups.

Anosov representations of amalgams

TL;DR

The paper develops a comprehensive framework for constructing Anosov representations of amalgams and HNN extensions of rank-one lattices into higher-rank Lie groups. Central to the approach is the Virtual Amalgam Theorem, which uses antipodal limit sets and interactive pairs/triples in flag varieties to realize amalgams as -Anosov subgroups after passing to suitable finite-index subgroups. It then applies faithful linear embeddings and exterior powers to promote -Anosov data to -Anosov representations, yielding new examples of one-ended hyperbolic groups that are linear and admit Anosov embeddings in higher rank but do not admit discrete faithful representations into any rank-one group. The paper also extends these methods to rank-one compatible pairs, divides subgroups, and automorphism fixed-point subgroups, producing a broad array of -Anosov doubles and HNN extensions, including new instances among , , , and . Overall, the work establishes a robust mechanism to generate higher-rank Anosov subgroups from rank-one lattices, enriching higher Teichmüller theory with numerous explicit constructions and applications.

Abstract

For uniform lattices in rank 1 Lie groups, we construct Anosov representations of virtual doubles of along certain quasiconvex subgroups. We also show that virtual HNN extensions of these lattices over some cyclic subgroups admit Anosov embeddings. In addition, we prove that for any Anosov subgroup of a real semisimple linear Lie group and any infinite abelian subgroup of , there exists a finite-index subgroup of containing such that the double admits an Anosov representation, thereby confirming a conjecture of [arXiv:2112.05574]. These results yield numerous examples of one-ended hyperbolic groups that do not admit discrete and faithful representations into rank 1 Lie groups but do admit Anosov embeddings into higher-rank Lie groups.
Paper Structure (24 sections, 41 theorems, 130 equations, 1 table)

This paper contains 24 sections, 41 theorems, 130 equations, 1 table.

Table of Contents

  1. Introduction
  2. Virtual amalgams
  3. Step 1: Construction of an interactive pair
  4. Step 2: Passing to finite index subgroups
  5. Proof of \ref{['thm:amalgam']}
  6. Amalgams of Anosov groups along abelian subgroups
  7. Faithful linear representations
  8. Positioning the limit sets
  9. Proof of \ref{['cyclic-amalgam']}
  10. Anosov amalgams along non-elementary subgroups
  11. Some preliminary results
  12. Proof of Theorem \ref{['amalgam-rank1']}
  13. Amalgams along dividing subgroups
  14. Amalgams along fixed-point subgroups of automorphisms
  15. Doubles of convex cocompact groups
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mathsf G$ and $\mathsf L$ be any pair of Lie groups listed in the same row of tab:compatible. Let $\Gamma$ be a convex cocompact group (for instance, a uniform lattice) in $\mathsf{G}$, and let $\mathrm H = \Gamma\cap \mathsf{L}$. Suppose further that $\mathrm H$ is a lattice in $\mathsf{L}$.

Theorems & Definitions (83)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 2.1: Virtual Amalgam Theorem
  • Remark 2.2
  • Theorem 2.3: Dey--Kapovich DeyK:amalgam
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 73 more