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2008 Tropicalization of group representations
Daniele Alessandrini
Algebr. Geom. Topol. 8(1): 279-307 (2008). DOI: 10.2140/agt.2008.8.279


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 π1(M) in SLn+1(). 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 π1(M) 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 SLn+1 to nonarchimedean 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 Teichmüller spaces. In the general case they are contractible metric spaces with a structure of tropical projective spaces.


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Daniele Alessandrini. "Tropicalization of group representations." Algebr. Geom. Topol. 8 (1) 279 - 307, 2008.


Received: 26 July 2007; Accepted: 20 November 2007; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1170.51005
MathSciNet: MR2443230
Digital Object Identifier: 10.2140/agt.2008.8.279

Primary: 51E24, 57M50, 57M60, 57N16

Rights: Copyright © 2008 Mathematical Sciences Publishers


Vol.8 • No. 1 • 2008
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