How frequent are discrete cyclic subgroups of semisimple Lie groups? (Q5928531)

Let \(G\) be a connected semisimple Lie group endowed with Haar measure \(\mu\). Denote by \(\Delta_G\) the set of all \(g\) \(G\) such that the cyclic subgroup generated by \(g\) is discrete. In a previous article (Generic subgroups of Lie groups, to appear), the author established that \(\mu(\Delta_G)=\mu(G \backslash \Delta_G)=\infty\). The present paper is concerned with the asymptotic behaviour of the ratio of volumes of the respective intersections with ''balls''. Here ''balls'' are defined in the following way. Let \(K\) be a maximal compact subgroup of \(G\) and consider \(X=K\backslash G/K\). Fix an exhaustion function \(\rho\) on \(X\), that is a continuous map from \(X\) to \(\mathbb R^+\) such that \(\rho^{-1}([0,r])\) is compact for all \(r\geq 0\). Consider \(\rho\) as a \(K\)-biinvariant exhaustion function on \(G\), and set \(B_r=\{g\in G: \rho(g)<r \}\). The main result is that the ratio \(\mu(\Delta_G\cap B_r)/\mu(B_r)\) tends to \(1\) when \(r\) tends to infinity.