Let $G=\text{Isom}(\mathbb{H}^n)$, $\Gamma<G$ be a lattice and endow $X=G/\Gamma$
with its $G$-invariant probability measure.

For a point $x\in X$ let $G_x$ be its stabilizer in $G$ (which is a lattice conjugated to $\Gamma$), let $\Lambda_x\subset \partial \mathbb{H}^n$ be the limit set of $G_x$ and let $C_x$ by the convex hull of $\Lambda_x$ in $\mathbb{H}^n$.
Clearly the maps $x\mapsto G_x,\Lambda_x,C_x$ are all $G$-equivariant.

Fix a base point $o\in\mathbb{H}^n$ and for every $p\in\mathbb{H}^n$ set
$$ f(p)=\int_X \left( d(p,C_x)-d(o,C_x)\right)dx. $$
Note that the integrand is bounded by $d(p,o)$ thus the integral is defined.
For every $p$,
$$ f(gp)=\int_X \left( d(gp,C_x)-d(o,C_x)\right)dx=
\int_X \left( d(p,C_{g^{-1}x})-d(g^{-1}o,C_{g^{-1}x})\right)dx=
\int_X \left( d(p,C_x)-d(g^{-1}o,C_x)\right)dx. $$
Specializing for $p=o$,
$$ f(go)=\int_X \left( d(o,C_x)-d(g^{-1}o,C_x)\right)dx, $$
and we get $f(gp)-f(go)=f(p)$.
Setting $\phi(g)=f(go)$ and subtituting $p=ho$ we get
$$\phi(gh)=f(gho)=f(ho)+f(go)=\phi(g)+\phi(h),$$
thus $\phi:G\to \mathbb{R}$ is a homomorphism. Since $G$ has no non-trivial homomorphism to $\mathbb{R}$, we conclude that for every $g\in G$, $\phi(g)=0$.
Equivalently, for every $p\in \mathbb{H}^n$, $f(p)=0$.
Since for every $x\in X$, $p\mapsto d(p,C_x)-d(o,C_x)$ is a convex function on $\mathbb{H}^n$ we conclude that for every $x$, $d(p,C_x)-d(o,C_x)$ is constant.
Substituting $p=o$ we get that this constant is $0$.
Taking $p\in C_x$ we get that $o\in C_x$ and conclude that for every $p$, $p\in C_x$.
That is $C_x=\mathbb{H}^n$. It follows that $\Lambda_x=\partial\mathbb{H}^n$.
Applying to the point $x$ with $G_x=\Gamma$ we get that the limit set of $\Gamma$ is $\partial\mathbb{H}^n$.