[Paper Review] On the SU(2,1) representation space of the Brieskorn homology spheres

Vu The Khoi|ArXiv.org|Nov 14, 2008

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper provides a trace coordinate parameterization of the SU(2,1) representation space for Brieskorn homology spheres, using traces of specific group elements and their commutators. It proves that irreducible representations are classified up to conjugation by the traces of $xy$, $x^{-1}y$, and the imaginary part of the trace of $[x,y]$, and applies this to show that the orbifold Toledo invariant fails to distinguish connected components in $\mathcal{R}^{*}_{\rm PU(2,1)}(\Sigma(2,3,13))$, despite detecting five components in $\Sigma(2,3,11)$.

ABSTRACT

In this paper, we give a parameterization of the SU(2,1) representation space of the Brieskorn homology spheres using the trace coordinates. As applications, we give an example which shows that the orbifold Toledo invariant in \cite{krebs} does not distinguish the connected components of the PU(2,1) representation space.

Motivation & Objective

• To develop a coordinate system for the SU(2,1) representation space of Brieskorn homology spheres using trace invariants.
• To classify irreducible representations up to conjugation via trace data of key group elements.
• To investigate the effectiveness of the orbifold Toledo invariant in distinguishing connected components of the PU(2,1) representation space.
• To provide explicit descriptions of the representation spaces for specific Brieskorn spheres, Σ(2,3,11) and Σ(2,3,13).
• To demonstrate that the Toledo invariant is insufficient to detect all connected components in certain cases, despite being a known lower bound tool.

Proposed method

• Derives trace identities for SU(2,1) matrices using invariant ring theory and algebraic relations in the free group of rank two.
• Applies trace identities to express the representation space in terms of coordinates: $t_{xy}$, $t_{x^{-1}y}$, and $\Im(t_{[x,y]})$.
• Uses the classification of conjugacy classes in SU(2,1) based on eigenvalue types (positive/negative) for elliptic elements.
• Applies the main theorem to compute explicit representations for Σ(2,3,11) and Σ(2,3,13), assuming $\rho(h) = I$.
• Performs computer-assisted searches to enumerate isolated irreducible representations and verify their distinctness in the PU(2,1) quotient space.
• Compares the number of connected components with the orbifold Toledo invariant values to assess its discriminating power.

Experimental results

Research questions

• RQ1 Can the SU(2,1) representation space of Brieskorn homology spheres be parameterized using trace coordinates of specific group elements and their commutators?
• RQ2 Are two irreducible SU(2,1) representations of a Brieskorn homology sphere conjugate if and only if the traces of $xy$, $x^{-1}y$, and the imaginary part of $[x,y]$ match?
• RQ3 Does the orbifold Toledo invariant fully distinguish the connected components of the PU(2,1) representation space for Brieskorn homology spheres?
• RQ4 How many irreducible representations exist in $\mathcal{R}^{*}_{\rm SU(2,1)}(\Sigma(2,3,11))$ and $\mathcal{R}^{*}_{\rm SU(2,1)}(\Sigma(2,3,13))$, and how do they relate to the Toledo invariant?
• RQ5 Can the Toledo invariant fail to distinguish connected components in the PU(2,1) representation space, even when it provides a lower bound?

Key findings

• The SU(2,1) representation space of Brieskorn homology spheres is fully parameterized by the traces $t_{xy}$, $t_{x^{-1}y}$, and $\Im(t_{[x,y]})$, with conjugacy determined by these invariants.
• For $\Sigma(2,3,11)$, there are exactly five irreducible representations in $\mathcal{R}^{*}_{\rm SU(2,1)}$, all with $\rho(h) = I$, and they yield five distinct points in $\mathcal{R}^{*}_{\rm PU(2,1)}$, matching the Toledo invariant lower bound.
• For $\Sigma(2,3,13)$, there are exactly eight isolated irreducible representations in $\mathcal{R}^{*}_{\rm SU(2,1)}$, all with $\rho(h) = I$, and they give eight distinct points in $\mathcal{R}^{*}_{\rm PU(2,1)}$.
• The orbifold Toledo invariant takes only seven distinct values for $\Sigma(2,3,13)$, despite eight connected components in the representation space, proving it does not distinguish all components.
• In $\Sigma(2,3,11)$, the Toledo invariant correctly detects five components, showing its bound is sharp in that case.
• The imaginary part of the commutator trace $\Im(t_{[x,y]})$ is zero for all listed representations, indicating that the commutator is elliptic with eigenvalues on the unit circle.