Relatively dominated representations from eigenvalue gaps and limit maps.

Area of Science:

• Group Theory
• Geometric Group Theory
• Topology

Background:

• Relatively dominated representations unify concepts like geometric finiteness (rank one) and Anosov condition (higher-rank analogue of convex cocompactness).
• Existing definitions often rely on quadratic gap assumptions.

Purpose of the Study:

• To refine the definition and understanding of relatively dominated representations.
• To establish new characterizations for these representations in higher-rank settings.
• To explore their connection with group actions and geometric properties.

Main Methods:

• Removing the quadratic gap assumption from the original definition.
• Developing characterizations based on eigenvalue gaps, analogous to existing results for Anosov representations.
• Formulating characterizations using singular value or eigenvalue gaps combined with limit maps.

Main Results:

• The quadratic gap assumption is successfully removed, broadening the applicability of the definition.
• A new characterization using eigenvalue gaps is provided, establishing a relative analogue for Anosov representations.
• Characterizations involving singular value/eigenvalue gaps and limit maps are formulated and applied.
• These characterizations are used to demonstrate that certain weak ping-pong group inclusion representations are relatively dominated.

Conclusions:

• Relatively dominated representations offer a unified framework for geometric finiteness and Anosov-type conditions.
• The removal of the quadratic gap assumption simplifies and extends the theory.
• New characterizations provide powerful tools for analyzing these representations and their associated group actions.
• The study connects abstract group theory with geometric properties through refined representation theory.