The case of equality in the dichotomy of Mohammadi–Oh

Abstract

If $n\ge 3$ and $\Gamma$ is a convex-cocompact Zariski-dense discrete subgroup of ${\mathbf{SO}}^o(1,n+1)$ such that ${\delta }_{\Gamma }=n-m$ where $m$ is an integer, $1\le m\le n-1$, we show that for any $m$-dimensional subgroup $U$ in the horospheric group $N$, the Burger–Roblin measure associated to $\Gamma$ on the quotient of the frame bundle is $U$-recurrent.