Let $X=\mathbb{H}^{n}/\Gamma$ be a quotient of hyperbolic space of a geometric finite subgroup. Let $\mu$ be the Bowen-Margulis measure on the unit tangent bundle, and $m$ its projection to $X$.

Fix a cusp $C$ of rank $r$, and let $C_{t}$ be the points at least $t$ deep in the cusp.

How does $m(C_{t})$ decay as $t\to \infty$?

Is it comparable to $e^{-rt}$?

I feel it should be possible to derive it from the Stratmann-Velani Global Measure Formula but am not sure how.


You can do it as an exercise, following the strategy in Dal'bo-Otal-Peigné "Série de Poincaré des groupes géométriquement finis", Israel J math. 118 (2000). See here, in french : http://www.lmpt.univ-tours.fr/~peigne/fichiers/israel.pdf

To compute the measure $\mu(T^1C_t)$ of your cusp, you find a nice fundamental domain for the action of the stabilizer $\Pi$ of your cusp on the universal cover, and your problem becomes to compute a sum over the elements $p\in \Pi$ such that the distance $d(o,p.o)$ is at least $2t$, which should, at the end, and if I did not do any mistake in a rapid computation, be comparable to something like $$\sum_{n\ge 2t} ne^{(\delta_\Pi-\delta_\Gamma)n}$$ , where $\delta_\Pi$ is the critical exponent of the parabolic subgroup associated to the cusp that you consider ($\delta_\Pi=r/2$ where $r$ is the rank of your cusp, and $\delta_\Gamma$ is the critical exponent of $\Gamma$.

This strategy does not use directly the Shadow Lemma (that you call Stratmann Velani Global measure formula, I guess), but the conformity properties of the Patterson measures at infinity.

If you need to read this kind of argument in english, we used it for a different purpose here (section 5.2) : http://www.lamfa.u-picardie.fr/schapira/recherche/distribution.pdf but the original (french) version of Dal'bo Otal Peigné is closer from what you need.


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