# Universal covering of a submanifold with boundary

Good afternoon,

I'm having a hard time trying to prove the following lemma :

Let $V$ be a hyperbolic compact orientable manifold, containing two totally geodesic com pact hypersurfaces $H$ and $G$ such that

1. $H$ and $G$ cuts $V$ into $2$ pieces

2. $H$ and $G$ intersects transversally at a codimesion $2$ manifold $W$ with angle $\frac{\pi}{2}$

This way V is divided into $4$ submanifolds with geodesic boundary, let $D$ be one of these submanifolds.

Let $\tilde{W}$ be a lift up of $W$ at the universal cover of $V$, which is $\mathbb{H}^n$, and let $\Gamma = \pi_1(D) \subset \text{Iso}(\mathbb{H}^n)$. Then $C=\text{Conv}(\Gamma \tilde{W})$ is a universal cover of $D$

I'm having trouble showing $C$ projects on $D$, even though it seems to be true when one draws a picture.

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