Benjamini-Schramm convergence of periodic orbits (Q2164373)

Let \(H\subset \mathrm{SL}_N({\mathbb R})\) be a non-compact semisimple group. Let \(\Delta_n\subset H\) be a sequence of lattices in \(H\). The sequence \(\{H/\Delta_n : n\in {\mathbb N}\}\) Benjamini-Schramm converges to \(H\) if for every \(r>0\) we have \[ \mu_n\left( \{y\in H/\Delta_n : \text{ injective radius of } y<r\}\right) \to 0 \] as \(n\to\infty\), where \(\mu_n\) denote the \(H\)-invariant probability measure on \( H/\Delta_n\) for every \(n\). The authors prove a criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups. This general observation is then applied to homogeneous spaces and the space of translation surfaces.