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.Comment: 27 pages, 1 figure; Changes in version 2: minor changes, some references added. Changes in version 3: the paper has been updated according to the companion paper arXiv:0801.0165 v1, some typos correcte

Convexity properties and complete hyperbolicity of Lempert's elliptic tubes

We prove that elliptic tubes over properly convex domains of the real projective space are C-convex and complete Kobayashi-hyperbolic. We also study a natural construction of complexification of convex real projective manifolds.Comment: 11 page

On the Tropicalization of the Hilbert Scheme

In this article we study the tropicalization of the Hilbert scheme and its suitability as a parameter space for tropical varieties. We prove that the points of the tropicalization of the Hilbert scheme have a tropical variety naturally associated to them. To prove this, we find a bound on the degree of the elements of a tropical basis of an ideal in terms of its Hilbert polynomial. As corollary, we prove that the set of tropical varieties defined over an algebraically closed valued field only depends on the characteristic pair of the field and the image group of the valuation. In conclusion, we examine some simple examples that suggest that the definition of tropical variety should include more structure than what is currently considered.Comment: 19 page

The behaviour of Fenchel-Nielsen distance under a change of pants decomposition

Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition $\mathcal{P}$ and given a base complex structure $X$ on $S$, there is an associated deformation space of complex structures on $S$, which we call the Fenchel-Nielsen Teichm\"uller space associated to the pair $(\mathcal{P},X)$. This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers \cite{ALPSS}, \cite{various} and \cite{local}, and we compared it to the classical Teichm\"uller metric (defined using quasi-conformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel-Nielsen metrics is not necessarily bi-Lipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal

On Fenchel-Nielsen coordinates on Teichm\"uller spaces of surfaces of infinite type

We introduce Fenchel-Nielsen coordinates on Teicm\"uller spaces of surfaces of infinite type. The definition is relative to a given pair of pants decomposition of the surface. We start by establishing conditions under which any pair of pants decomposition on a hyperbolic surface of infinite type can be turned into a geometric decomposition, that is, a decomposition into hyperbolic pairs of pants. This is expressed in terms of a condition we introduce and which we call Nielsen convexity. This condition is related to Nielsen cores of Fuchsian groups. We use this to define the Fenchel-Nielsen Teichm\"uller space associated to a geometric pair of pants decomposition. We study a metric on such a Teichm\"uller space, and we compare it to the quasiconformal Teichm\"uller space, equipped with the Teichm\"uller metric. We study conditions under which there is an equality between these Teichm\"uller spaces and we study topological and metric properties of the identity map when this map exists

Compattificazione di varietà di caratteri e applicazioni topologiche

Versione italiana L'argomento della tesi è lo studio dei caratteri di rappresentazioni del gruppo fondamentale di una varietà in SL_2(C) con lo scopo di ottenere delle informazioni sulla topologia della varietà. Considero una varietà M e definisco R(M) come l'insieme delle rappresentazioni del suo grupp fondamentale in SL_2(C). L'insieme dei caratteri delle rappresentazioni in R(M) sarà chiamato X(M). Se il gruppo è finitamente generato, su R(M) e X(M) si può dare una struttura di varietà algebrica affine (cosa non banale per X(M) ), sfruttando la quale si definisce il concetto di punto "all'infinito" di X(M), e si associano a questi punti una valutazione di un campo F e una rappresentazione del gruppo in SL_2(F). A partire da questi elementi si costruisce un'azione del gruppo su un albero reale, un particolare tipo di spazio metrico che nel caso in cui la valutazione è a valori in Z è un albero simpliciale. Queste azioni su alberi danno informazioni sulla topologia di M. Si affronta in maniera particolareggiata il caso dell'azione su un'albero simpliciale, facendo vedere come, se M è una 3-varietà compatta, si può arrivare a costruire un sistema di superfici incompressibili di M. Si fa vedere come con opportune ipotesi su M si possono ottenere superfici incompressibili con certe proprietà, e si mostrano alcune applicazioni, come ad esempio la dimostrare un bel risultato di decomposizione di una 3-varietà. Nel caso di una varietà associata ad un nodo si riesce a costruire un invariante del nodo, l' A-polinomio, e si calcolano esplicitamete le pendenze di bordo delle superfici incompressibili costruite in questo modo. Il caso delle azioni su alberi reali viene presentato per mostrare una nuova costruzione del bordo di Thurston degli spazi di Teichmuller ed una nuova interpretazione dei punti di bordo aggiunti. English version The aim of this thesis is studying how the characters of SL_2(C)-representations of the fundamental group of a manifold may bring topological information about the manifold itself. Let M be a manifold, R(M) be the set of SL_2(C)-representations of its fundamental group, X(M) be the set of characters of such representations. If the fundamental group is finitely generated we can endow R(M) and X(M) with an algebraic structure (which is not simple for X(M)), and using this structure we can define the "points at infinity" of X(M), and we can associate with each of these points a valutation of a field F and a representation of the group in SL_2(F). Starting from these elements we can construct an action of the group on a real tree, a particular kind of metric space that, if the valutation is Z-valued, is an ordinar simplicial tree. These actions bring information about the topology of M. I have studied in detail the actions on simplicial trees, and I have shown that, if M is a compact 3-manifold, an action on a simplicial tree permits the construction of a system of incompressible surfaces in M. With additional hypotheses on M, it is possible to obtain incompressible surfaces with some additional properties, and I have shown some applications of this, as for example a decomposition result for 3-manifolds. If M is a knot manifold, it is possible to construct a knot invariant, the A-polynomial, and it is possible to evaluate the boundary slopes of incompressible surfaces constructed this way. The actions on real trees lead to a new construction of the Thurston boundary of Teichmuller spaces and to a new interpretation of the boundary points. <BR