Good afternoon,
Take a submanifold $V$ of codimension $1$ of the sphere at infinity of $\mathbb{H}^n$ which is not the sphere at infinity of a totally geodesic hyperplane $\mathbb{H}^{n-1} \subset \mathbb{H}^n$. Now suppose that $f$ is an isometry of $\mathbb{H}^n$ fixing $V$ pointwise. Is true that $f$ must be the identity ?
Thank you for your answers
Yes: you can easily see that $f$ must fix some point $x$ in the convex hull of the set $V$ (for exemple the center of an ideal triangle with vertices in $V$), and thus it fixes pointwise every geodesic from $x$ to a point of $V$; by your hypothesis on $V$ the union of all these geodesics is not contained in a proper totally geodesic submanifold of $\mathbb{H}^n$ and (as the set of fixed points of an isometry is totally geodesic) it follows that $f$ must be the identity on $\mathbb{H}^n$.
