A Basmajian-type inequality for the indefinite orthogonal group.

The purpose of this work is to prove a Basmajian-type inequality for the indefinite orthogonal group $O(2,n)$. If you are wondering what Basmajian- type means, then probably you do not know the Basmajian identity: this beautiful equality was published by Basmajian in 1993 ([Bas93]) and it al- lows us to compute the length of the boundary of a compact hyperbolic surface in function of the length of some particular kind of geodesic con- tained in this surface.

The Basmajian-type inequality proved in this thesis is, instead, a gener- alization working in the context of the Hermitian symmetric space associated to the Lie group $S{O}_0(2,n)$, for $n\ge 3$.

Before of starting with the proper work, let me explain more in details what this Basmajian identity states and why one should consider exactly $S{O}_0(2,n)$ for a possible generalization.