Self-Joinings of cusped quasi-Fuchsian manifolds: local mixing, coding and counting

We study the dynamics of discrete subgroups arising from products of representations of surface groups. Let $S$ be an oriented punctured hyperbolic surface of finite area. We consider a $k$-tuple of distinct points $\rho_1, \dots, \rho_k$ in the Quasi-Fuchsian space $\text{QF}(S)$ of discrete, faithful, type-preserving, geometrically finite representations of $\pi_1(S)$ into $\text{PSL}_2(\mathbb{C})$. The diagonal product $\rho = \prod_{i=1}^{k} \rho_i$ defines a self-joining subgroup $\Gamma_\rho =\rho(\pi_1(S))$ of $G = \prod_{i=1}^{k} \text{PSL}_2(\mathbb{C})$. 

We establish local mixing for the diagonal flow on $\Gamma_\rho \backslash G$, obtain precise asymptotics for matrix coefficients, and derive orbital counting and equidistribution results for the simultaneous action of $\pi_1(S)$ on $\mathbb{H}^3 \times \dots \times \mathbb{H}^3$. To handle the presence of cusps, we construct an infinite countable Markov coding and analyze the associated transfer operators given by vector-valued cocycles. All results extend to a more general setting of $k$-tuples of pairwise non-conjugate, type-preserving geometrically finite representations of $\pi_1(S)$ into $\text{SO}(n_i, 1)$ with $n_i \ge 2$. This is a work in progress with Dongryul Kim and Hee Oh.