Primitive stability and the Bowditch conditions revisited

TL;DR

This paper revisits and simplifies the proof of the equivalence between primitive stability and Bowditch conditions for $SL(2,\mathbb{C})$ representations of the free group $F_2$, using the Bowditch tree and ideas from Lee and Xu.

Contribution

The paper provides a simplified proof of the equivalence between primitive stability and BQ-conditions, integrating Lee and Xu's ideas and the Bowditch tree framework.

Findings

Proved the equivalence of primitive stability and BQ-conditions.

Simplified the original proof using new ideas and the Bowditch tree.

Enhanced understanding of $SL(2,\mathbb{C})$ representations of $F_2$.

Abstract

The equivalence of two conditions on the primitive elements in an $SL(2,\mathbb C)$ representation of the free group $F_2 = <a,b>$, namely Minsky's condition of primitive stability and the $BQ$-conditions introduced by Bowditch and generalised by Tan, Wong and Zhang, has been proved by Lee and Xu and independently by the author in arXiv:1901.01396. This note is a revised version of our original proof, which is greatly simplified by incorporating some of the ideas introduced by Lee and Xu, combined with the language of the Bowditch tree.