• Carnot groups
• finite perimeter
• geometric measure theory

In Chapter 1 we present some recent results of Geometric Measure Theory in doubling metric measure spaces and in Ahlfors k-reguar spaces, we define the class of functions of Bounded Variation and the sets of finite perimeter, following the the paper of Miranda [6]. In the second section we state some results of the theory of finite perimeter sets in Ahlfors spaces, contained in [1]. Those theorems will play and important role when dealing with Carnot groups, that are a particular example of Ahlfors spaces. In the last section we focus our attention to the Special functions of Bounded Variation, we prove some theorems on the structure of the discontinuous set of a BV function and we define the U-spaces.
In Chapter 2 we state the main features abount Carnot-Carathéodory spaces. In section (2.1) we recall the definition of Lie group and Lie algebra and we prove the existence of a diffeomorphism between a nilpotent Lie group and its Lie algebra. In section (2.2) we give the definition of CC-distance, we state the Theorem of Chow and, following [3], we analize the structure of the tangent set to a CC-space. In section (2.3) we prove the existence (see [5]) of the minimal upper gradient of a continuous function in CC-spaces, continuous with respect to the Euclidean distance.
Chapter\ 3 is the core of the thesis, here we begin giving the definition of a Carnot group and proving some basic properties of the CC-distance associated to a basis of the first layer of the Lie algebra. Section (3.2) is devoted to the exposition of the results contained in [2], where it is proved the existence of a vertical halfspaces in the tangent set to a Carnot group. In section (3.3) we generalize some method of [2] to prove the locality property for every Carnot group. Finally, in section (3.4) we present the rectifiability problem in Carnot groups and we state the main results of [4] .

Bibliography
[1] L. Ambrosio, Some fine properties of set of finite perimeter in Ahlfors regular metric spaces; Advanced in Math. 159,(2001) 51-67 .
[2] L. Ambrosio, B. Kleiner and E. Le Donne, Rectifiability of Sets of Finite Perimeter in Carnot Groups: Existence fo a Tangent Hyperplane; preprint.
[3] A. Bella\"iche, Sub-Riemannian Geometry; in sub-Riemannian Geometry, Progress in Mathematics, 144, edited by A. Bella\"iche and J.-J. Risler, Birkh\"auser Verlag, Basel 1996.
[4]B. Franchi, R. Serapioni and F. S. Cassano, On the structure of Finite Perimeter Sets in Step 2 Carnot Groups; J. Geom. Analysis, 13 (2003), 421-466.
[5] P. Hajlaz, P. Koskela, Sobolev met Poincar\'e.
[6] M. Miranda Jr., Functions of bounded variation on ``good'' metric spaces; J. Math. Pures Appl. 82 (2003) 975-1004.