# Cusps in hyperbolic manifolds and fundamental group

I am reading the book "The Arithmetic of Hyperbolic Three Manifolds" by Maclachlan and Reid and I am having some problems in understanding something about cusps.

The definition they give of a cusp is the following (1.2.7):

A point $\zeta \in \overline{\mathbb{C}}$, the sphere at infinity, is a cusp of the Kleinian group $\Gamma$ if the stabiliser $\Gamma_\zeta$ contains a free abelian group of rank 2.

This is ok for me. Anyway, I know another definition of cusp (or, better, of "cusp neighborhood"), the one coming from the thick-thin decomposition. Obviously, one cannot expect these to be the same thing, but my claim would be the following:

Let $M= \frac{\mathbb{H}^3}{\Gamma}$ be a hyperbolic (finite volume, complete, connected, orientable) 3-manifold. There is a bijection between the cusps under the action af the group $\Gamma$ (that is, $\zeta \cong \zeta'$ if exists $A \in \Gamma$ s.t. $A\zeta = \zeta'$) and the cusp neighborhoods obtained in the thick-thin decomposition.

I have another problem related with this is much more specific, and I think it should be easier, but I find some difficulties. From some consideration made in the rest of the book, i would believe this, but I cannot prove it.

In the same settings as above, let $T_i \times [0, \infty)$ be a cusp neighborhood. Let $\zeta_i$ be the corresponding cusp in $\overline{\mathbb{C}}$. Then $\pi_1(T_i \times \{0 \})$ embeds in $\Gamma$ through $i^*$ (the inclusion) and its image is (after composing, if necessary, with conjugation) $\Gamma_{\zeta_i}$.

Is this true? Can you give me some reference?