Algebraic & Geometric Topology

Tropicalization of group representations

Daniele Alessandrini

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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.

Article information

Algebr. Geom. Topol., Volume 8, Number 1 (2008), 279-307.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M60: Group actions in low dimensions 57M50: Geometric structures on low-dimensional manifolds 51E24: Buildings and the geometry of diagrams 57N16: Geometric structures on manifolds [See also 57M50]

projective structure Bruhat–Tits building tropical geometry character representation


Alessandrini, Daniele. Tropicalization of group representations. Algebr. Geom. Topol. 8 (2008), no. 1, 279--307. doi:10.2140/agt.2008.8.279.

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