Jordan and Cartan spectra in higher rank with applications to correlations

TL;DR

This paper investigates the asymptotic distribution of length and displacement spectra for higher rank representations of groups, establishing precise growth rates and correlations using Jordan and Cartan projections within Anosov subgroups.

Contribution

It introduces a new framework for analyzing correlations of spectra in higher rank groups via Jordan and Cartan projections, extending previous results to tubes and interior vectors.

Findings

Established exponential growth rates for spectral correlations.

Proved invariance of growth indicators between Jordan and Cartan projections.

Extended spectral distribution results to tubes in higher rank symmetric spaces.

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 $\varepsilon_1, \dots, \varepsilon_d>0$, there exist $c_1,c_2> 0$ such that \begin{align*} &\#\{[\gamma]\in [\Gamma]: v_iT \le \ell_{\rho_i(\gamma)} \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_1 \frac{e^{\delta_\rho (\mathsf{v})T}}{ T^{{(d+1)}/{2}}};\\ &\#\{\gamma\in \Gamma: v_iT \le \mathsf{d}(\rho_i(\gamma)o,o) \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_2 \frac{e^{\delta_\rho (\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.