A note on degenerations of Morse actions

TL;DR

This paper investigates Morse representations of discrete groups in higher rank Lie groups, revealing that unbounded sequences of such representations impose specific structural constraints on the groups involved.

Contribution

It establishes a connection between unbounded Morse representations and the structural properties of the underlying groups, extending understanding of their degenerations.

Findings

Unbounded Morse representations imply specific structural properties of the groups.

Sequences of Morse representations can be characterized by their boundedness in the character variety.

The results provide insights into the degeneration behavior of Morse actions in higher rank Lie groups.

Abstract

We study Morse representations of discrete subgroups in higher rank semi-simple Lie groups defined by M. Kapovich, B. Leeb and J. Porti. We show that, if a sequence of Morse representations $\rho_n : \Gamma \rightarrow G$ is (strongly) unbounded in the character variety, the group must have a very particular structure.