When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi\_1M$ is relatively hyperbolic relative to the parabolic groups $\pi\_1\partial M$, so one can consider $\partial\Gamma$ which is defined as the ideal boundary of the coned off Cayley-Graph (i.e. the Cayley-Graph of $\pi\_1M$ coned off over the right cosets of $\pi\_1\partial M$).


Question: Is $\partial\Gamma$ homeomorphic to the (n-1)-sphere and, if yes, where has this been proved?


In the closed case this is immediate from the quasiisometry between $\Gamma$ and the universal covering $\widetilde{M}$. In the cusped case of course $\Gamma$ is quasiisometric to the complement of the horoballs in $\widetilde{M}$, but the cones over the parabolic subgroups seem not to be quasiisometric to the horoballs, so it is not clear to me how to adapt the argument.



Boundaries of relatively hyperbolic groups
As pointed out in the comments, the boundary of the
coned-off graph won't be a sphere in this case. In fact, it won't
even be compact. 


Bowditch's paper on relatively hyperbolic groups (recently published
in
[*IJAC*](http://www.worldscientific.com/doi/abs/10.1142/S0218196712500166))
gives some details about the relationship between the two boundaries
in the last few sections. In particular see Propositions 8.5 and 9.1.


To summarize Bowditch's results: If $\hat{\Gamma}$ is the coned-off
Cayley graph for a relatively hyperbolic pair, and $B$ is the standard
boundary for the relatively hyperbolic pair, then
$\partial\hat{\Gamma}$ sits inside $B$ as a dense subset. The
complement is a countable (also dense) set which can be identified
with the set of maximal parabolic subgroups. 


In the case you are
interested in $B$ is an $(n-1)$-sphere. Thus $\partial\hat{\Gamma}$
is an $(n-1)$-sphere with a countable dense subset removed.


(EDIT: I was going to add this as a comment, but I don't have enough
reputation. Let $X$ be the space obtained by coning off the boundary
of $M$, and let $Y$ be the universal cover of $X$. The space $Y$ isn't
quasi-isometric to $\hat{\Gamma}$. Rather the normal closure $N$ of
the parabolic subgroups acts on $\hat{\Gamma}$, with a quotient
quasi-isometric to $Y$.)


