| In this thesis, we mainly consider two problems: the enumeration of lattices and the entropy rigidity in the automorphism groups of trees and buildings. For the first problem, we obtain a universal upper bound for the number un(Gamma) of lattices containing a fixed lattice Gamma with index n, in the automorphism groups of trees and simply connected polyhedral complexes. We also give a lower bound of u n(Gamma) for certain types of uniform lattices in the automorphism groups of trees and some hyperbolic buildings.;The second problem is related to the dynamics of the geodesic flow on trees and buildings. Among the normalized metrics on a graph, we show the existence and the uniqueness of an entropy-minimizing metric, and give explicit formulae for the minimal volume entropy and the metric realizing it. |
