Precise local mixing for self-joinings of cusped surfaces via infinite coding

Let $S$ be an oriented punctured hyperbolic surface of finite area. We consider $k$-tuples $\rho_1, \dots, \rho_k$ of pairwise non-conjugate, type-preserving, geometrically finite representations of $\pi_1(S)$ into $\SO(n_i,1)$ for $n_i \geq 2$. The diagonal product $\rho = \prod_{i=1}^k \rho_i$ defines a self-joining subgroup $\Gamma_\rho = \rho(\pi_1(S))$ of $G = \prod_{i=1}^k \SO(n_i,1)$. We establish a precise local mixing result for the diagonal flow on $\Gamma_\rho \backslash G$, obtaining an asymptotic expansion of every order for the correlation function. To handle cusps, our proof proceeds in two parts. Geometrically, we construct an infinite countable Markov coding. Analytically, we establish the spectral properties of the associated transfer operators, which are given by vector-valued cocycles and act on appropriate Banach spaces. This is a joint work with Dongryul Kim and Hee Oh.

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