arXiv subjects

Daniele Alessandrini

Publications and source records attributed to Daniele Alessandrini.

21 records · Page 2Linked to original sources

arXiv2007

A compactification for the spaces of convex projective structures on manifolds

In this paper we construct a compactification for the parameter space of convex projective structures on a fixed n-manifold M. This parameter space is a closed semi-algebraic subset of the variety of characters of representations of the fundamental group of M in SL_{n+1}(R). The boundary is the inverse limit of an inverse system of logarithmic limit sets of this semi-algebraic set, in a sense it is the tropicalization of the parameter space. The interpretation of the boundary points can also be given using tropical geometry. This construction is a generalization of the construction of compactification of the Teichmüller spaces.

math.GT↗

arXiv2007

Tropicalization of group representations

In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an n-manifold M. These spaces are closed semi-algebraic subsets of the variety of characters of representations of the fundamental group of M in SL_{n+1}(R). The boundary was constructed as the tropicalization of this semi-algebraic set. Here we show that the geometric interpretation for the points of the boundary can be constructed searching for a tropical analogue to an action of the group on a projective space. To do this we need to construct a tropical projective space with many invertible projective maps. We achieve this using a generalization of the Bruhat-Tits buildings for SL_{n+1} to non-archimedean fields with real surjective valuation. In the case n = 1 these objects are the real trees used by Morgan and Shalen to describe the boundary points for the Teichmuller spaces. In the general case they are contractible metric spaces with a structure of tropical projective spaces.

math.GT↗

arXiv2005

Amoebas, tropical varieties and compactification of Teichmuller spaces

In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via tropical varieties. When we apply this construction to the Teichmuller spaces we see that they can be mapped in a real algebraic hypersurface in such a way that the cone over the boundary is a subpolyhedron of a tropical hypersurface. We want to show how some properties of the boundary becomes straightforward if looked at from this point of view. For example there is a PL structure that appears naturally on the boundary, and it is shown to be equivalent to the one defined by Thurston. Also we may see easily that every polynomial relation among trace functions on Teichmuller space may be turned automatically in a tropical relation among intersection forms over the boundary.

math.AG↗