Convergence and counting in infinite measure

We construct non-uniform convergent lattices of pinched, negatively curved Hadamard spaces, in any dimension N > 2. These lattices are exotic, by which we mean that they have a maximal parabolic subgroup P < such that (P) = ().We also give examples of divergent, non-uniform exotic lattices in dimension N = 2. Finally, we consider a particular class of such exotic lattices, with infinite Bowen–Margulis measure and whose cusps have a particular asymptotic profile (satisfying a “heavy tail condition”), and we give precise estimates of their orbital function; namely, we show that their orbital function is lower exponential with asymptotic behaviour eR R1−L(R) , for a slowly varying function L.