## Final Activity Report Summary - GEOMGROUPSMFLD (Geometry of groups and manifolds)

The project studied geometric properties of mathematical objects like groups and manifolds. Roughly speaking, our objects are infinite graphs constructed recursively with the same basic pieces (like an infinite network.) We were interested in global properties and large scale geometries.

The mathematical procedure followed to do that is similar to a backward zoom, where one looks from far and far away an infinite object; losing in that way local particulars but obtaining an accurate asymptotic global picture. The most interesting fact is that from the global picture we can still recover local information. For instance, one of the main objectives achieved in the project is that symmetric spaces are completely characterised by their asymptotic pictures.

Another research line of the project studied the possible deformations of our object when a supplementary dimension is added to the universe where they live. Here, we established rigidity results for a certain class of hyperbolic manifolds: no deformations are possible for them. In a suitable sense this can be viewed as the impossibility to deform a certain class of fractals of the plane to fractals of the space.

The third main result of GEOMGROUPMFLD is the study of geodesics in our strange graphs. What is the shortest railways-path form Strasbourg to Marseille? In that casi it is probably through Paris, but providing an exact (mathematically speaking) answer is a difficult problem in general. In particular, we studied metric properties and shortest paths in the so-called Outer Space, with particular attention to the relations between geodesics and a class of known paths, called folding paths, showing that folding paths are almost geodesics.

The mathematical procedure followed to do that is similar to a backward zoom, where one looks from far and far away an infinite object; losing in that way local particulars but obtaining an accurate asymptotic global picture. The most interesting fact is that from the global picture we can still recover local information. For instance, one of the main objectives achieved in the project is that symmetric spaces are completely characterised by their asymptotic pictures.

Another research line of the project studied the possible deformations of our object when a supplementary dimension is added to the universe where they live. Here, we established rigidity results for a certain class of hyperbolic manifolds: no deformations are possible for them. In a suitable sense this can be viewed as the impossibility to deform a certain class of fractals of the plane to fractals of the space.

The third main result of GEOMGROUPMFLD is the study of geodesics in our strange graphs. What is the shortest railways-path form Strasbourg to Marseille? In that casi it is probably through Paris, but providing an exact (mathematically speaking) answer is a difficult problem in general. In particular, we studied metric properties and shortest paths in the so-called Outer Space, with particular attention to the relations between geodesics and a class of known paths, called folding paths, showing that folding paths are almost geodesics.