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Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space

David Xu

Abstract

We construct convex-cocompact representations of fundamental groups of closed hyperbolic surfaces into the isometry group of the infinite-dimensional hyperbolic space using bendings. We prove that convex-cocompact representations of finitely generated groups in the group of isometries of the infinite-dimensional hyperbolic space form an open set in the space of representations and that the space of deformations (up to conjugation) obtained by bending an irreducible representation of a surface group is infinite-dimensional.

Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space

Abstract

We construct convex-cocompact representations of fundamental groups of closed hyperbolic surfaces into the isometry group of the infinite-dimensional hyperbolic space using bendings. We prove that convex-cocompact representations of finitely generated groups in the group of isometries of the infinite-dimensional hyperbolic space form an open set in the space of representations and that the space of deformations (up to conjugation) obtained by bending an irreducible representation of a surface group is infinite-dimensional.
Paper Structure (13 sections, 32 theorems, 18 equations)

This paper contains 13 sections, 32 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. Preliminaries
  4. The infinite-dimensional hyperbolic space
  5. Discreteness of subgroups
  6. Stability of convex-cocompact representations into Isom(Hinfty)
  7. Convex-cocompact representations and quasi-isometric embeddings
  8. Stability of convex-cocompact representations
  9. Convex-coboundedness and convex-cocompactness
  10. Embeddings of Isom(H2) into Isom(Hinfty)
  11. A density theorem for convex-cocompact subgroups
  12. Spectral representation for real Hilbert spaces
  13. Bending irreducible representations from Isom(H2)

Key Result

Theorem 1.1

Let $\Gamma$ be a finitely generated group and $\rho : \Gamma \to \mathop{\mathrm{Isom}}\nolimits(\mathbf{H}^{\infty})$ be a representation. The two following assertions are equivalent.

Theorems & Definitions (42)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 2.1: DSU, 5.2.1 and 5.2.3
  • Remark 2.2
  • Proposition 2.3: DSU, Propositions 5.2.4 and 5.2.7
  • Definition 3.1
  • Proposition 3.2: GH, Proposition 7.14
  • Theorem 3.3: DSU, Theorem $12.2.12$
  • ...and 32 more