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For a convex cocompact  Kleinian group $\Gamma <\rm{SO}(n,1)$,  Sullivan (around 1985) established a fundamental relation among the critical exponent, the bottom of the L^2-spectrum of  the hyperbolic manifold $\Gamma\backslash {\mathbb H}^n$,  the quasi-regular representation $L^2(\Gamma\backslash G)$ and the Hausdorff dimension of the limit set.  We consider a higher rank analogue of this relation. For self-joinings of convex cocompact Kleinian groups (or more generally for any Anosov subgroup of a product of rank one simple algebraic groups), we discover a surprising fact that they satisfy  a similar relation as convex cocompact groups with “small” critical exponents.

This talk is based on joint works with Dongryul Kim and Yair Minsky, and with Sam Edwards in different parts.