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On the domination of surface-group representations in $\mathrm{PU}(2,1)$

Pabitra Barman, Krishnendu Gongopadhyay

TL;DR

This work addresses domination of surface-group representations into ${ m PU}(2,1)$ for punctured surfaces by establishing a dominant real Fuchsian representation. The authors exploit ${ t Z}$-invariants attached to adjacent real ideal triangles in a fixed ideal triangulation to parametrize $T$-bent representations and then construct a dominant representation $ ho_0: abla_1(S_{g,k}) ightarrow { m PO}(2,1)$ by taking absolute values of edge invariants, with WillBending providing discreteness and faithfulness. They prove that $ ho_0$ dominates $ ho$ in the Bergman translation-length spectrum, preserving peripheral loop lengths, via a positive-expression framework that yields spectral radius inequalities $ abla abla( ho( abla(\gamma))) abla abla( ho_0( abla(\gamma)))$ for all $ abla(\gamma)$. The results extend domination phenomena known for ${ m PSL}_2(f C)$ to ${ m PU}(2,1)$ in the punctured-surface setting, introduce the bending-fiber viewpoint for ${ m PU}(2,1)$-representations, and connect modulus of traces to spectral domination, contributing to the broader program of Higher Teichmüller theory in complex hyperbolic geometry.

Abstract

This article explores surface-group representations into the complex hyperbolic group $\mathrm{PU}(2,1)$ and presents domination results for a special class of representations called $T$-bent representations. Let $S_{g,k}$ be a punctured surface of negative Euler characteristic. We prove that for a $T$-bent representation $ρ: π_1(S_{g,k}) \rightarrow \mathrm{PU}(2,1)$, there exists a discrete and faithful representation $ρ_0: π_1(S_{g,k}) \rightarrow \mathrm{PO}(2,1)$ that dominates $ρ$ in the Bergman translation length spectrum, while preserving the lengths of the peripheral loops.

On the domination of surface-group representations in $\mathrm{PU}(2,1)$

TL;DR

This work addresses domination of surface-group representations into for punctured surfaces by establishing a dominant real Fuchsian representation. The authors exploit -invariants attached to adjacent real ideal triangles in a fixed ideal triangulation to parametrize -bent representations and then construct a dominant representation by taking absolute values of edge invariants, with WillBending providing discreteness and faithfulness. They prove that dominates in the Bergman translation-length spectrum, preserving peripheral loop lengths, via a positive-expression framework that yields spectral radius inequalities for all . The results extend domination phenomena known for to in the punctured-surface setting, introduce the bending-fiber viewpoint for -representations, and connect modulus of traces to spectral domination, contributing to the broader program of Higher Teichmüller theory in complex hyperbolic geometry.

Abstract

This article explores surface-group representations into the complex hyperbolic group and presents domination results for a special class of representations called -bent representations. Let be a punctured surface of negative Euler characteristic. We prove that for a -bent representation , there exists a discrete and faithful representation that dominates in the Bergman translation length spectrum, while preserving the lengths of the peripheral loops.

Paper Structure

This paper contains 6 sections, 16 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex hyperbolic plane and its isometries
  4. Totally geodesic subspaces and ideal triangles
  5. The $T$-bent representations and their ${\tt Z}$-invariants
  6. Proof of the theorem

Key Result

Theorem 1.3

Let $S_{g,k}$ be an oriented surface of negative Euler characteristic with at least one puncture. For a $T$-bent representation $\rho:\pi_1(S_{g,k}) \rightarrow \mathrm{PU}(2,1)$, there exists a discrete and faithful representation $\rho_0:\pi_1(S_{g,k}) \rightarrow \mathrm{PO}(2,1)$ that dominates

Figures (4)

  • Figure 1: $M_z$flips the vertices of the adjacent real ideal triangles.
  • Figure 2: Part of the modified dual graph (bold).
  • Figure 3: Computing $\rho(\gamma)$.
  • Figure 4: Segment of a lift of a peripheral loop $\gamma$.

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4: Bending fiber for $\mathrm{PU}(2,1)$-representations
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1
  • Theorem 2.1
  • Definition 2.2: Cartan invariant
  • Proposition 2.3
  • ...and 25 more