# The case of equality in the dichotomy of Mohammadi-Oh

## TL;DR

This paper investigates the recurrence properties of Burger-Roblin measures for certain convex-cocompact Zariski-dense subgroups of SO(1,n+1), revealing a specific case of measure recurrence related to the group's critical exponent.

## Contribution

It establishes that for subgroups with a critical exponent equal to an integer minus a dimension, the Burger-Roblin measure is recurrent along any m-dimensional subgroup in the horospheric group.

## Key findings

- Burger-Roblin measure is U-recurrent for specific subgroups
- Recurrence depends on the critical exponent being an integer minus a dimension
- Results connect geometric group properties with measure-theoretic recurrence

## Abstract

If $n \geq 3$ 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, $1 \leq m \leq n-1$, 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.