Dynamics on the SU(2,1)-character variety of the one-holed torus

TL;DR

The article investigates SU(2,1) representations of the once-punctured torus group and their moduli via the SU(2,1) character variety, introducing a Bowditch-type domain of discontinuity $ abla X_{BQ}$ that strictly contains the convex cocompact locus. By leveraging Lawton’s explicit SL(3,C)–character-variety parametrization and Bowditch-style combinatorics on the one-holed-torus complexes, the authors characterize Bowditch representations with equivalent conditions, establish the openness of $ abla X_{BQ}$, and prove proper discontinuity of the mapping class group action. A key methodological contribution is the construction of an attracting subgraph $T_ ho(K)$ governed by trace growth (Fibonacci growth) and the demonstration that Bowditch representations yield finite attracting graphs for all $K>M(c)$, ensuring dynamical tameness beyond convex cocpacity. Topological results show the SU(2,1) character variety $ abla X(F_2, ext{SU}(2,1))$ deformation retracts onto a torus $S^1 imes S^1$, highlighting nontrivial topology distinct from the $ ext{SU}(3)$ case. Overall, the work extends primitive-stable/Bowditch-type domains to complex hyperbolic geometry, offering computable criteria for domain membership and a framework for analyzing mapping class group dynamics on relative character varieties.

Abstract

We study the relative SU(2,1)-character varieties of the one-holed torus, and the action of the mapping class group on them. We use an explicit description of the character variety of the free group of rank two in SU(2,1) in terms of traces, which allow us to describe the topology of the character variety. We then combine this description with a generalization of the Farey graph adapted to this new combinatorial setting, using ideas introduced by Bowditch. Using these tools, we can describe an open domain of discontinuity for the action of the mapping class group which strictly contains the set of convex cocompact characters, and we give several characterizations of representations in this domain.

Figures (7)

• Figure 1: The complexes $\mathcal{C}$ (in grey) and $\mathcal{T}$ (in black), and the labeling of the connected components of $\mathbf{H}^2 \setminus \mathcal{T}$ once we fix the generators of $F_2 = \langle \alpha, \beta \rangle$.
• Figure 2: Oriented edge $(X, Y, Z; T \longrightarrow T')$. If you forget the orientation, the same red edge can be denoted $(X; Y, Z)$.
• Figure 3: The complex $\mathcal{E}$ with the coloring of its edges and regions of ${\mathbf H}^2 \setminus \mathcal{T}$.
• Figure 4: $f(t)=0$
• Figure 5: A vertex which is a fork (left) and one which is not (right). The two arrows pointing away from the vertex have to be in different triangles to get a fork.

Theorems & Definitions (69)

• Theorem A
• Theorem B
• Theorem C
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Theorem 2.6: Theorem 6.2.4, gol-chyp