Jordan and Cartan spectra in higher rank with applications to correlations

Abstract

For a given $d$-tuple $\rho =({\rho }_1,\dots ,{\rho }_d):\Gamma \to G$ of faithful Zariski dense convex cocompact representations of a finitely generated group $\Gamma$, we study the correlations of length spectra $\{{\ell }_{{\rho }_i(\gamma )}{\}}_{[\gamma ]\in [\Gamma ]}$ and correlations of displacement spectra $\{\mathsf{d}({\rho }_i(\gamma )o,o){\}}_{\gamma \in \Gamma }$. We prove that for any interior vector $\mathsf{v}=(v_1,\dots ,v_d)$ in the {\it{spectrum cone}}, there exists ${\delta }_{\rho }(\mathsf{v})>0$ such that for any ${\epsilon }_1,\dots ,{\epsilon }_d>0$, there exist $c_1,c_2>0$ such that \begin{align*} &\#\{[γ]\in [Γ]: v_iT \le \ell_{ρ_i(γ)} \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_1 \frac{e^{δ_ρ(\mathsf{v})T}}{ T^{{(d+1)}/{2}}};\\ &\#\{γ\in Γ: v_iT \le \mathsf{d}(ρ_i(γ)o,o) \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_2 \frac{e^{δ_ρ(\mathsf v)T}}{ T^{{(d-1)}/{2}}}. \end{align*} We deduce this result as a special case of our main theorem on the distribution of Jordan projections with holonomies and Cartan projections {\it{in tubes}} of an Anosov subgroup $\Gamma$ of a semisimple real algebraic group $G$. We also show that the growth indicator of $\Gamma$ remains the same when we use Jordan projections instead of Cartan projections and tubes instead of cones, except possibly on the boundary of the limit cone.

46 pages, 4 figures, Final version, To appear in Moduli